Introduction To Hypergeometric, Supertrigonometric, And Superhyperbolic Functions 9780128241547

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An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions

An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions Xiao-Jun Yang State Key Laboratory for Geomechanics and Deep Underground Engineering School of Mathematics China University of Mining and Technology Xuzhou, Jiangsu, China

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2021 Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-12-824154-7 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Mara Conner Acquisitions Editor: Chris Katsaropoulos Editorial Project Manager: Aleksandra Packowska Production Project Manager: Anitha Sivaraj Designer: Victoria Pearson Esser Typeset by VTeX

To my family, parents, brother, sister, wife, and my daughters

Contents Biography Preface 1. Euler gamma function, Pochhammer symbols and Euler beta function

xi xiii

1

1.1. Euler gamma function

1

1.2. Pochhammer symbols

6

1.3. Euler beta function

9

2. Hypergeometric, supertrigonometric, and superhyperbolic functions via Clausen hypergeometric series

13

2.1. Clausen hypergeometric series

13

2.2. The hypergeometric supertrigonometric functions via Clausen hypergeometric series

49

2.3. The hypergeometric superhyperbolic functions via Clausen hypergeometric series

73

2.4. The special functions via Clausen hypergeometric series with three numerator parameters and two denominator parameters

97

2.5. Analytic number theory via Clausen hypergeometric functions

115

3. Hypergeometric supertrigonometric and superhyperbolic functions via Gauss hypergeometric series

139

3.1. Gauss hypergeometric series

139

3.2. Hypergeometric supertrigonometric functions via Gauss hypergeometric series

145

3.3. Hypergeometric superhyperbolic functions via Gauss hypergeometric series

149 vii

viii

Contents

3.4. Some elementary examples for the Gauss hypergeometric series

154

3.5. Integral representations for the hypergeometric superhyperbolic and hypergeometric superhyperbolic functions

169

3.6. Analytic number theory via Gauss hypergeometric functions

203

4. Hypergeometric supertrigonometric and superhyperbolic functions via Kummer confluent hypergeometric series

237

4.1. The Kummer confluent hypergeometric series of first type

237

4.2. The hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of first type

242

4.3. The hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of first type

254

4.4. The Kummer confluent hypergeometric series of second type

265

4.5. The hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of second type

267

4.6. The hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of second type

276

4.7. Analytic number theory via Kummer confluent hypergeometric series

284

5. Hypergeometric supertrigonometric and superhyperbolic functions via Jacobi polynomials

293

5.1. Jacobi polynomials

293

5.2. Jacobi–Luke polynomials

309

5.3. Jacobi–Luke-type polynomials

312

6. Hypergeometric supertrigonometric and superhyperbolic functions via Laguerre polynomials

345

6.1. Laguerre polynomials

345

6.2. Extended works containing the Laguerre polynomials

390

6.3. Some results based on the special functions

417

Contents

7. Hypergeometric supertrigonometric and superhyperbolic functions via Legendre polynomials

ix

445

7.1. Legendre polynomials

445

7.2. Legendre-type polynomials

456

References

471

Index

481

Biography About the Author Xiao-Jun Yang PhD, is a full-time professor of Applied Mathematics and Mechanics at China University of Mining and Technology, Xuzhou, China. His scientific interests include Viscoelasticity, Mathematical Physics, Fractional Calculus and Applications, Fractals, Analytic Number Theory, and Special Functions. He was awarded the Obada-Prize, Cairo, Egypt (2019). He was a recipient of the Young Scientist Award (2019) for the contributions in developing the Local Fractional Calculus at ICCMAS-2019, Istanbul, Turkey, and the Springer Distinguished Researcher Award at ICMMAAC-2019, Jaipur, India. He is currently an Editor or Associate Editor of several scientific journals including Applied Numerical Mathematics, Mathematical Methods in the Applied Sciences, Fractals, Mathematical Modelling and Analysis, Journal of Thermal Stresses, International Journal of Geometric Methods in Modern Physics, Alexandria Engineering Journal, and Advances in Difference Equations. He is the Highly Cited Researcher (2020 and 2019, Clarivate Analytics) in Mathematics, and Elsevier Most Cited Chinese Researcher in Mathematics (2017, 2018, and 2019). He is a member of the Scientific Committee of 10th edition of the Pan African Congress of Mathematicians. He is the author and coauthor of six monographs for Elsevier, CRC, World Science, and Asian Academic and coeditor of one edited book in De Gruyter.

xi

Preface The theory of the special functions plays an important role in the study of the mathematical models of the real-world phenomena with use of the functions in the fields of mathematics, physics, chemistry, engineering, and other applied sciences. The main aim of the monograph is not only to explain the role of the hypergeometric series in modern pure and applied mathematics but also to give the audience an idea for the structure of the new special functions. It is designed for professional mathematicians, physicists, engineers, chemists, and graduate students in those and closely allied fields who have had no previous knowledge of the theory of the hypergeometric, supertrigonometric, and superhyperbolic functions. Due to the above-mentioned topics of the special functions containing the Clausen hypergeometric series, Gauss hypergeometric series, Kummer confluent hypergeometric series, Jacobi polynomials, Laguerre polynomials, and Legendre polynomials, we demonstrate a great many of the identities for the special functions and take a systematic account of the supertrigonometric and superhyperbolic functions from the viewpoint of the Euler’s tasks. The monograph is divided into seven chapters as follows. Chapter 1 provides the elements of the gamma function, Pochhammer symbols and beta functions. The Weierstrassian product, Legendre duplication formula, Weierstrass theorem, Winckler theorem, Stirling theorem, Hankel integral theorem, and Gauss multiplication formula are presented in detail. The incomplete gamma functions, incomplete beta function, and the logarithmic derivative of the gamma function are also considered. Chapter 2 is devoted to the hypergeometric supertrigonometric functions and the hypergeometric superhyperbolic functions via Clausen hypergeometric series. The Laplace transforms for the supertrigonometric and superhyperbolic functions, the zeros of the supertrigonometric and superhyperbolic functions via Clausen hypergeometric series, and the identities of the supertrigonometric and superhyperbolic functions via Clausen hypergeometric series are considered in detail. The analytic number theory involving the above special functions are also discussed. Chapter 3 gives an account of the hypergeometric supertrigonometric and superhyperbolic functions via Gauss hypergeometric series. The Laplace transform, Chu– Vandermonde identity, Gauss differential equation, Mellin transform, Gauss theorem, Stirling theorem, Pfaff theorem, Koshliakov theorem, and Bateman theorem are considered in detail. The analytic number theory invoking the hypergeometric supertrigonometric and superhyperbolic functions via Gauss hypergeometric series are also presented based on the proposed results. xiii

xiv

Preface

Chapter 4 proposes the hypergeometric supertrigonometric and superhyperbolic functions via the Kummer confluent hypergeometric series. The analytic number theory for the hypergeometric, supertrigonometric, and superhyperbolic functions via Kummer confluent hypergeometric series are suggested in detail. Chapter 5 addresses the definitions, properties, and theorems for the Jacobi, Jacobi– Luke, and Jacobi–Luke-type polynomials. The hypergeometric supertrigonometric functions and hypergeometric superhyperbolic functions via Jacobi and Jacobi–Luke-type polynomials and their generating functions are discussed. The Laplace transforms and integral representations for the new special functions are also addressed. Chapter 6 introduces the concepts, properties, and theorems of the Laguerre polynomials and illustrates the supertrigonometric and superhyperbolic functions containing the Laguerre polynomials via Laguerre polynomials, Szegö function of first type, Rainville function, and Szegö function of second type. We also report the Brafman theorem, Hille theorem, Feldheim theorem, Weisner theorem, and other theorems for new special functions. Chapter 7 presents the definitions, properties, and theorems for the Legendre polynomials and Legendre-type polynomials. We propose the supertrigonometric functions and superhyperbolic functions via Legendre polynomials and Legendre-type polynomials. Professor Dr. Xiao-Jun Yang would like to express grateful thanks to Professor George E. Andrews, Professor H.M. Srivastava, Professor Bo-Ming Yu, Professor Wolfgang Sprößig, Professor Souza De Cursi Eduardo, Professor Nouzha El Yacoubi, Professor Mourad Ismail, Professor Tom H. Koornwinder, Professor Michel L. Lapidus, Professor Simeon Oka, Professor Roland W. Lewis, Professor Manuel Lopez Pellicer, Professor Michael Reissig, Professor George A. Anastassiou, Professor Yalchin Efendiev, Professor Alain Miranville, Professor Mahmoud Abdel-Aty, Professor Stefano Galatolo, Professor Dumitru Mihalache, Professor Martin Bohner, Professor Thiab Taha, Professor Chin-Hong Park, Professor Sung Yell Song, Professor Qin Sheng, Professor Salvatore Capozziello, Professor André Keller, Professor Martin Ostoja-Starzewski, Professor Minvydas Ragulskis, Professor J.A. Tenreiro Machado, Professor Mauro Bologna, Professor Carlo Cattani, Professor Semyon Yakubovich, Associate Professor Jian-Hua Li, Ms. Karin Uhlemann, Professor Ayman S. Abdel-Khalik, Professor Mircea Merca, Professor Irene Maria Sabadini, Professor Mario Di Paola, Professor Mokhtar Kirane, Professor Giuseppe Failla, Professor Reza Saadati, Professor Amar Debbouche, Professor Yusif Gasimov, and Professor Ivanka Stamova. By way of acknowledgment, we are grateful to Professor Zhi-Ming Ye, Professor WeiYi Su, Professor Zhi-Ying Wen, Professor Jin-De Cao, Professor Mao-An Han, Professor Wei Lin, Professor Yi Wang, Professor Long Jiang, Professor Hui-Lai Li, Professor SanYang Liu, Professor Wen-Xia Li, Professor Sheng-Bo Li, Professor Qi-Gui Yang, Professor Qing Qian, Professor Heng-Shan Hu, Professor Xi-Lin Xie, Professor Yi Cheng, Professor Jian-Jun Zhang, Professor Mei-Qian Chen, Professor Fang Li, Professor Zhi-Liang Zhu, Professor Shu-Kai Duan, Professor Yong-Gui Kao, Professor Jing-Yue Xue, Professor Zhen Jin, Professor Yu-Feng Zhang, Professor Ai-Ming Yang, Professor Zhao-Jun Ou-Yang, Professor

Preface

xv

Jun-Guo Lu, Professor Yuan Cao, Professor Wei-Qiu Chen, Professor Hui-Ming Wang, Professor Guo-Tao Wang, Professor Wei Jiang, Professor Fa-Zhan Geng, Professor Bao-Li Xie, Professor Wen-Bin Liu, Professor Deng-Yin Wang, Professor Xin-An Ren, Professor LianYing Miao, Professor Xiang-Juan Yao, Professor Gang Wu, Professor Hu Shao, Professor Sheng-Jun Fan, Professor Hui-Xing Zhang, Professor Bo Wu, Professor Xing-Jie Yan, and Professor Jian-Hua Yang. My special thanks go to Professor He-Ping Xie, Professor Feng Gao, Professor Cheng-Bin Wu, Professor Guo-Qing Zhou, Professor Fu-Bao Zhou, Professor Hong-Wen Jing, Professor Yang Ju, Professor Hong-Wei Zhou, Professor Tian-Jian Zhou, Professor Wei Lai, Professor Dong Ding, Professor Ming-Zhong Gao, Professor Xian-Biao Mao, Professor Zhan-Qing Chen, Professor Hai-Bo Bai, Professor Zhan-Guo Ma, Professor Hai Pu, Dr. Pei-Tao Qiu, my PhD students Jian-Gen Liu and Yi-Ying Feng, and the financial support of the YueQi Scholar of the China University of Mining and Technology (Grant No. 102504180004). Finally, I also wish to express my special thanks to Elsevier staff, especially to C. Katsaropoulos, M. Conner, A. Packowska and A. Sivaraj for their cooperation in the production process of this book. Xiao-Jun Yang State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, China May 27, 2020

1 Euler gamma function, Pochhammer symbols and Euler beta function 1.1 Euler gamma function In this section, we introduce the history and concept of the Euler gamma function, give the theorems for the Euler gamma function, and investigate the definition for the incomplete Euler gamma function.

1.1.1 Definition and theorems for the Euler gamma function Let C, R, Z, and N be the sets of complex numbers, real numbers, integers, and natural numbers, respectively. Let Z+ , R+ , Z− , and R− be the sets of the positive integers, positive real numbers, negative integers, and negative real numbers. − Let Z− 0 = Z ∪ 0 and N0 = N ∪ 0. Let Re (x) denote the real part of x ∈ C. Definition 1.1 (Euler integral of first kind). [Euler (1729)] The gamma function due to Euler is defined as [1]  ∞  (z) = e−t t z−1 dt

(1.1)

0

for z ∈ C with Re (z) > 0. The formula (called the Euler integral of first kind) was discovered by Euler in 1729 (see [1], p. 1, [2]), and the notation  (z) was introduced by Legendre in 1814 (see [3], p. 476). Theorem 1.1 (Weierstrass product). [Weierstrass (1856)] The gamma function can be expressed as [4]  (z) =

 ∞  e−γ z   z −1 z 1+ ek , z k

(1.2)

k=1

where z ∈ C\Z− 0 , and γ := limn→∞

 n

1 k=1 k

 − log n is the Euler constant. Moreover,  (z) is

analytic except at the points z ∈ Z− 0 , where it has simple poles [5]. An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions https://doi.org/10.1016/B978-0-12-824154-7.00007-6 Copyright © 2021 Elsevier Inc. All rights reserved.

1

2

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

The formula for the Weierstrass product was discovered by Weierstrass in 1856 [4] and by Newman in 1848 [6], and the proofs were published by Hölder (1886) [7], Moore (1896) [8], and Baines (1900) [9]. Definition 1.2 (Euler’s functional equation). [Euler (1729)] The Euler’s functional equation states  (z + 1) = z (z)

(1.3)

for z ∈ C with Re (z) > 0. Euler’s functional equation was discovered by Euler in 1729 [2] and reported by Weierstrass (1856) [4], Brunel (1886) [10], Gronwall (1916) [11], and Olver (1997) [12]. Theorem 1.2 (Euler theorem). [Euler (1729)]  (z + 1) = z!

for z ∈ N0 .

(1.4)

Euler’s functional equation was discovered by Euler in 1729 [1,2] and discussed by Weierstrass (1856) [4], Brunel (1886) [10], and Gronwall (1916) [11]. Theorem 1.3 (Euler theorem). [Euler (1729)]   √ 1  = π. 2

(1.5)

This equality was discovered by Euler in 1729 [1] and discussed by Bell (1968) [13], Luke (1969) [14], and Bendersky (1933) [15]. Theorem 1.4 (Euler theorem). [Euler] n 

 

j =1

1−j n

 = (2π)

n−1 2

1

n2

for n, j ∈ N.

(1.6)

The Euler theorem was reviewed by Gronwall in 1916 [11]. Theorem 1.5 (Winckler theorem). [Winckler (1856)]  n−1  hj  m ngz+ mn−m−n  mz + j =0 n−m n 2 = (2π) 2 ,  m−1

nl n l=0  nz + m

(1.7)

where z ∈ C and k, j, k, m, n ∈ N. The Winckler theorem was discovered by Winckler in 1856 [16] and reviewed by Gronwall in 1916 [11].

Chapter 1 • Euler gamma function, Pochhammer symbols and Euler beta function 3

Theorem 1.6. [Schlömilch (1844) and Newman (1848)] ∞   z z 1 = eγ z z 1 + e− k  (z) k

for z ∈ C and k ∈ N.

(1.8)

k=1

The result was discovered by Schlömilch in 1844 [17] and by Newman in 1848 [6]. Theorem 1.7 (Whittaker theorem). [Whittaker (1902)]  ∞  (z) e−kt t z−1 dt = z k 0

(1.9)

for z ∈ C with Re (z) > 0 and k ∈ N. The Whittaker theorem was first reported by Whittaker in 1902 (see [18], p. 184) and further reported by Whittaker and Watson in 1920 [19]. Theorem 1.8. [Whittaker (1902)]

   π 2 1  α2  α2 α−1 β−1   cos t sin tdt = 2  α+β 0

(1.10)

2

for Re (α) > 0 and Re (β) > 0. The result was first obtained by Whittaker in 1902 (see [18], p. 191) and further reported by Whittaker and Watson in 1920 [19]. Theorem 1.9. [Titchmarsh (1948)] If α + β > 1, then  ∞ −∞

2α+β−1 1 dt = .  (α + t)  (β − t)  (α + β − 1)

(1.11)

The result was first reported in Titchmarsh’s monograph [20]. Theorem 1.10. [Titchmarsh (1948)] ∞   (α + 1)  (β + 1) k (α + β + k) , =  (α + β + 1) (α + k) (β + k)

(1.12)

k=1

provided that Re (α) > −1, Re (β) > −1, and Re (α + β) > −1. The result was first presented in Titchmarsh’s monograph [20]. Theorem 1.11. [Titchmarsh (1948)] n

n   −1    1 z 2πi k−1 n zn 1− n = −  −e k k=1

k=1

for z ∈ C and k, n ∈ N.

(1.13)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

The result was first reported in Titchmarsh’s monograph [20]. Theorem 1.12 (Euler’s completion formula). [Euler (1771)]  (z)  (1 − z) = and sin (πz) = πz

π sin (πz)

 ∞   z2 1− 2 , k

(1.14)

(1.15)

k=1

where Re (z) > 0 and z ∈ C. The formula is due to Euler (1771) [21]. For more detail of the results, we refer the readers to Weierstrass (1856) [4], Manocha and Srivastava (1984) [22], Luke (1969) [14], Bell (1968) [13], Godefroy (1903) [23], and Tannery (1910) [24]. Theorem 1.13 (Legendre duplication formula). [Legendre (1809), p. 485]     1 1  (2z)  = 22z−1  (z)  z + for z ∈ C\Z− 0. 2 2

(1.16)

Legendre’s duplication formula was first discovered by Legendre in 1809 (see [3], p. 477). For more detail on the Legendre duplication formula, we refer the readers to Gronwall (1916) [11], Andrews et al. (1999) [25], and Manocha and Srivastava (1984) [22]. Theorem 1.14 (Gauss multiplication theorem). [Gauss (1812)]  (mz) = (2π)

1−m 2

1

mmz− 2

m  j =1

  j −1  z+ m

(1.17)

  j for z ∈ C\ 0, − m with j < m and j, m ∈ N. Gauss’ multiplication formula is due to Guass (1812) [26]. For more detail on Gauss’ multiplication formula, we refer the readers refer to Winckler (1856) [16], Gronwall (1916) [11], Manocha and Srivastava (1984) [22], and Andrews et al. (1999) [25]. Theorem 1.15 (Weierstrass theorem). [Weierstrass (1856)]     1 π 2π 1 −z  + z = πsec (πz) = =  2 2 cos (πz) eiπz + e−iπz

(1.18)

for z ∈ C/Z− 0. The result was discovered by Weierstrass (1856) [4] and reported by Bell in 1968 [13] and by Luke in 1969 [14].

Chapter 1 • Euler gamma function, Pochhammer symbols and Euler beta function 5

Theorem 1.16. We have     1 1 π 2π  + iz  − iz = = , 2 2 cosh (πz) eπz + e−πz   2   2π  1 + iz  = ,   πz 2 e + e−πz

(1.19) (1.20)

and 2π π  = πz −iz sin (πzi) z e − e−πz

 (iz)  (−iz) =

(1.21)

as |z| → ∞. The results were reported by different researchers, for example, Lerch (1893) [27], Godefroy (1901) [28], Stieltjes (1889) [29], Bateman (1955) [30], and Andrews et al. (1999) [25]. Theorem 1.17. We have n−1  j =1

    j j (2π )n−1  1− = ,  n n n

  n √π 1 n 2 = (−1)  −n + , 2 (2n − 1)! √   1 (2n − 1)! π =  n+ , 2 2n  n−1   πz z2 2 − 1)!) ,  1 − ((n sin (πz) j

 (n + z)  (n − z) =

(1.22)

(1.23) (1.24)

(1.25)

j =1

and

  2 1       n  n +  2 1 1 4z2  n+ +z  n+ −z =  1− 2 2 cos (πz) (2j − 1)2 j =1

(1.26)

for n ∈ N and z ∈ C\Z− 0. The results were reported by Weierstrass in (1856) [4] and Wang et al. (1979) (see [31], pp. 588–589). Theorem 1.18 (Incomplete gamma functions). [Schlömilch (1871)] The incomplete gamma functions are defined as  − (a, z) =

0

a

e−t t z−1 dt (|arg (a)| < π)

(1.27)

6

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and

 + (a, z) =

a

e−t t z−1 dt (|arg (a)| < π)

(1.28)

0

with  (z) = − (a, z) + + (a, z) ,

(1.29)

where a > 0 and z ∈ C with Re (z) > 0. These functions are introduced and called “incomplete gamma functions” by Schlömilch (1871) ([32]; also see [33]) and further reported by J. Tannery (1882) [34], Prym (1877) [35], Lerch (1905) [36], and Thomson (1947) [37]. Theorem 1.19 (Pearson theorem). [Pearson (1934)] We have − (a, z) = z−1 a z e−a 1 F1 (1; z + 1; a)

(1.30)

for z ∈ Z− 0. The result was discovered by Pearson [38] and reported by Slater (1966) [39], Wang et al. (1979) [31], and Manocha and Srivastava (1984) [22]. Theorem 1.20. Let h(1) (t) > 0, h (0) = 0, h (∞) = ∞, and x ∈ C with Re (x) > 0. Then we have  1  (x) = e−h(t) (h (t))x−1 h(1) (t) dt.

(1.31)

0

The result was discovered by Yang, Gao, and Ju [40] in 2020 for x ∈ N. Theorem 1.21. Let x ∈ C with Re (x) > 0. Then we have  (x) =



0

−1

e−e etx dt, t

(1.32)

where e−e is the Euler function. t

For more detail of the incomplete gamma function, see Pearson (1957) [38], Haskins (1915) [41], and Pagurova (1963) [42].

1.2 Pochhammer symbols In this section, we introduce the Pochhammer symbols and give theorems on their properties.

Chapter 1 • Euler gamma function, Pochhammer symbols and Euler beta function 7

1.2.1 Definition and theorems for the Pochhammer symbols Definition 1.3 (Pochhammer symbol). [Pochhammer (1870)] The Pochhammer symbol is defined as [43] (α)k =

k 

(α + n − 1)

n=1

= =

(1.33)

 (α + k)  (α)  1 (k = 0) , α (α + 1) · · · (α + k − 1) (k ∈ N0 ) ,

and (α)0 = 1,

(1.34)

where α ∈ C and k, n ∈ N. The Pochhammer symbol was first suggested and used by Pochhammer in 1870 [43]. Weierstrass [4] noticed in 1856 that  (α + k) = α (α + 1) · · · (α + k − 1)  (α) (k ∈ N0 ) .

(1.35)

For more information, see the monograph [44]. Moreover (see [4,11,39]), lim (α)k =

k→∞

1  (α)

for α ∈ C\C− 0 and k ∈ N. Suppose that α = −n and n ∈ N0 . Then we have (see [39], p. 3)  (−n)k , n ≥ k, (α)k = −, n < k.

(1.36)

(1.37)

Theorem 1.22 (Euler limit theorem). [Euler (1729)] We have nz n→∞ (z)n+1

 (z) = lim

(1.38)

for z ∈ C\Z− 0. The result was discovered by Euler in 1729 [2], reported by Weierstrass in 1856 [4], and discussed by Gronwall in 1916 [11].

8

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 1.23. We have the identities (α)k (α + k)n = (α)n+k

(1.39)

and (α + k)m−k =

(α)m (α)k

(1.40)

for α ∈ C\Z− 0 and n, k ∈ N. The first formula was reported by Rainville in 1960 (see [45], p. 59), and the second formula was suggested by Slater in 1966 (see [39], p. 31). Theorem 1.24. We have (1)n = n!,

  α n α (α − 1) · · · (α − k − 1) = n!  (α + 1) = n! (α − n + 1) 1 = n! (α + 1)n (−1)n (−α)n , = n!  (α + 1) = (−1)n (−α)n ,  (α − n + 1)

(1.41)

(1.42)

(1.43)

1 (−1)n (−m)n , = m! (m − n)!

(1.44)

 (α − n) (−1)n = ,  (α) (1 − α)n

(1.45)

 (α − n) (−1)n = (α)−n = ,  (α) (1 − α)n

(1.46)

n! n! (n + 1)! − = , (α)n+1 (α + 1)n+1 (α)n+2

(1.47)

(α)n (α)n (α)n+1 − = (α − β) , (β)n (β)n+1 (β)n+1

(1.48)

(α)n−k =

(−1)k (−1)k (α)n = , (1 − α)−n (1 − α − n)k (1 − α − n)k

(1.49)

Chapter 1 • Euler gamma function, Pochhammer symbols and Euler beta function 9

(−1)k (1)n (−1)k n! = , (−n)k (−n)k α  α + 1 , (α)2n = 22n 2 n 2 n  m   α+j −1 mn , (α)mn = m m n

(1)n−k = (n − k)! =

(1.50) (1.51) (1.52)

j =1

and

 (−n)k =

(−1)k n! (n−k)!

(0 ≤ k ≤ n)

(1.53)

0 (k > n)

for j, k, m, n ∈ N0 , k ≤ n, and α, β ∈ C\Z. For more detail, see [4,11,13,14,22,39]. Theorem 1.25 (Stirling theorem). [Stirling (1730)] 1 z−α ∞  (α)k = (z)k+1 k=0

∞   (z)  (α + k) =  (α)  (z + k + 1)

(1.54)

k=0

=

α α (α + 1) 1 + + + ··· , z z (z + 1) z (z + 1) (z + 2)

for α, z ∈ C\Z− 0 with Re (α) > 0, Re (z) > 0, and Re (α − z) > 0. The result was discovered by Stirling in 1730 [46] and reviewed by Gronwall in 1916 [11].

1.3 Euler beta function In this section, we give the definition of the Euler beta function, present theorems on the Euler beta function, and discuss the definition of the incomplete Euler beta function.

1.3.1 Definition and theorems for the Euler beta function Definition 1.4 (Euler beta function). [Euler (1772)] The Euler beta function is defined as  1  (α)  (β) B (α, β) = = t α−1 (1 − t)β−1 dt  (α + β) 0 for α, β ∈ C\Z− 0 with Re (α) > 0 and Re (β) > 0.

(1.55)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

The formula (called the Euler integral of the second kind) was first discovered by Euler in 1772 [47] and by Legendre in 1811 (see [48], p. 221); the name of the beta function was introduced by Binet in 1839 [49]. For more detail, see the monograph [44]. Theorem 1.26. [Whittaker and Watson (1902)] We have  (α, n) = lim nα B (α, n) n→∞

(1.56)

for α ∈ C\Z− 0 and n ∈ N. The results were reported by Whittaker and Watson (see [19], p. 254). Theorem 1.27. [Euler (1772)] We have B (α, β) (α + β, c) = B (β, c) (β + c, α) .

(1.57)

The result was discovered by Euler in 1772 [47] and further reported by Whittaker and Watson (see [19], p. 261). Theorem 1.28. [Whittaker and Watson (1902)]  ∞ t α−1 (1 + t)−(α+β) dt, B (α, β) =

(1.58)

0

B (α, n + 1) =

n! (n ∈ N0 ) , (α)n+1

(1.59)

β B (α, β) , α+β

(1.60)

B (α, β + 1) =

B (α, β) = B (α + 1, β) + B (α, β + 1) , B (α, β + 1) =

β B (α + 1, β) , α

(1.61) (1.62)

and B (α, β) = B (β, α) .

(1.63)

The results were discovered by Whittaker and Watson (see [19], p. 254). Theorem 1.29. [Whittaker and Watson (1902) [19]] We have   n n−1 B β + , α k=1 n B (nα, nβ) = n−nβ n B − 1) α, α) ((k k=1 for n ∈ N and α, β ∈ C\Z− 0 with Re (α) > 0 and Re (β) > 0.

(1.64)

Chapter 1 • Euler gamma function, Pochhammer symbols and Euler beta function 11

Definition 1.5 (Incomplete beta function). [Pearson (1934)] The incomplete beta function is defined as [50]  z zα Bz (α, β) = t α−1 (1 − t)β−1 dt = 2 F1 (α, 1 − β; α + 1; z) . α 0

(1.65)

The incomplete beta function was introduced by Pearson in 1934 [50] and further reported by Thomson (1947) [37] and Srivastava and Kashyap (1982) [51]. For more detail and history, see Dutka (1981) [52]. Theorem 1.30 (Hankel integral theorem). [Hankel (1864)] We gave the identity [53]  1 1 = s −z es ds, s ∈ C,  (z) 2πi L

(1.66)

where L is the Hankel contour. The result was discovered by Hankel in 1864 [53]. There is an alternative integral representation as follows. Theorem 1.31. [Hankel (1864)] i 1 =  (z) 2π

 ℵ

(−s)−z e−s ds, s ∈ C,

(1.67)

where ℵ is the loop contour starting at 0i + ∞, encircling the origin, and tending to 0i − ∞. The result was discovered by Hankel in 1864 [53]. Definition 1.6 (The logarithmic derivative of the gamma function). The logarithmic derivative of the gamma function is defined as [11] ψ (z) = for s ∈ C and

d log  (z)  (1) (z) = dz  (z)

  √ 1 log  (z) = z − log z − s + log 2π +  (z) 2

(1.68)

(1.69)

with the infinite series  (z) due to Gudermann (1845) [54]. Theorem 1.32. Let h(1) (t) > 0, h (1) = 1, h (∞) = ∞, and x, y ∈ C with Re (x) > 0 and Re (y) > 0. Then we have  ∞ B (x, y) = e−h(t) (h (t))x−1 (1 − h (t))y−1 h(1) (t) dt. (1.70) 0

The result was discovered by Yang, Gao, and Ju in 2020 for x ∈ R+ and y ∈ R+ [40].

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 1.33. Let x, y ∈ C with Re (x) > 0 and Re (y) > 0. Then we have  ∞

y−1 t e−e et 1 − et dt, B (x, y) = −1

(1.71)

where e−e is the Euler function. t

For more information of the Euler gamma function, Pochhammer symbols, and Euler beta function, we refer the reader to Gronwall (1916) [11], Bell (1968) [13], Luke (1969) [14], Whittaker and Watson [19], Manocha and Srivastava (1984) [22], Wang et al. (1979) [31], Slater [39], Rainville [45], and Andrews et al. (1999) [25].

2 Hypergeometric, supertrigonometric, and superhyperbolic functions via Clausen hypergeometric series 2.1 Clausen hypergeometric series In 1828, Clausen [55] introduced the hypergeometric series, which is now called the generalized hypergeometric series or Clausen hypergeometric series (also called the Clausen hypergeometric function). In fact, Wallis [56] first used in 1656 the term hypergeometric to denote the series to make it beyond the ordinary geometric series at that time. After that, Euler studied similar series in 1748 [57]. Vandermonde extended the binomial theorem in 1770 [58]. Gauss introduced the famous Gauss hypergeometric series in 1812 [26]. On the consideration of the Gauss hypergeometric series, Clausen [55] proposed the Clausen hypergeometric series.

2.1.1 Definition, convergence, and properties for the Clausen hypergeometric series Let us start with the definition, properties, and theorems for the Clausen hypergeometric series. Definition 2.1. [Clausen (1828)] The Clausen hypergeometric series are defined as [55] ((a) , (c) ; z)   = p Fq a 1 , · · · , ap ; c1 , · · · , cq ; z   a , · · · , ap = p Fq 1 ;z c1 , · · · , cq   ∞  (a1 )n · · · ap zn  n , = (c1 )n · · · cq n n!

p Fq

(2.1)

n=0

where an , cn , z ∈ C and n, p, q ∈ N0 . This extended version of the Gauss hypergeometric series was introduced by Clausen in 1828 [55]. The results were developed by Dutka (1984) [59], Bailey (1935) [60], Karlsson An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions https://doi.org/10.1016/B978-0-12-824154-7.00008-8 Copyright © 2021 Elsevier Inc. All rights reserved.

13

14

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

(1995) [61], Opps et al. (2005) [62], Miller and Paris (2012) [63], Aomoto et al. (2011) [64], Roy (1987) [65], Nilsson (2009) [66], Wilson (1978) [67], Fine (1988) [68], Gasper et al. (2004) [69], Andrews (1974) [70], and Natanzon (1979) [71]. The formula was systematically studied by Thomae (1870) [72] and Appell and De Fériet (1926) [73]. Theorem 2.1 (Convergence). [60] The cases of convergence of the Clausen hypergeometric series (2.1) for an ∈ C\Z− 0: 1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges for |z|  < 1, diverges for |z| > 1, and for |z| = 1, it q absolutely p converges absolutely if Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0. The result was reported by, for example, Bailey (1935) [60], Srivastava and Kashyap (1982) [51], Slater (1966) [39], Andrews et al. (1999) [25], and Rainville (1960) [45]. We now provide some properties of the Clausen hypergeometric series. Theorem 2.2 (The differential equation for the Clausen hypergeometric series). If an , cn , z ∈ C, n, p, q ∈ N0 , and |z| < 1, then the hypergeometric series (2.1) is a solution of the differential equation (Q (q, μ, cn ) ϕ) (z) − (P (p, μ, an ) ϕ) (z) = 0,

(2.2)

(Q (q, μ, cn ) ϕ) (z)   q    d d  ϕ (z) + (cn − 1) ϕ (z) z = z dz dz n=1 q   

d d  z + (cn − 1) ϕ (z) =z dz dz

(2.3)

(P (p, μ, an ) ϕ) (z)  p   dϕ (z) z + an ϕ (z) =z dz n=1   p   d =z z + an ϕ (z). dz

(2.4)

where

n=1

and

n=1

The result was reported by some researchers, for example, Srivastava and Kashyap (1982) [51], Rainville (1960) [45], Andrews et al. (1999) [25], Luke (1969) [14], Nørlund (1955) [74], Smith [75,76], Mehlenbacher (1938) [77], Littlejohn and Kanwal (1987) [78], Dwork (1984) [79], Zarzo and Dehesa (1994) [80], Takemura (2012) [81], and Plastino and Rocca (2015) [82].

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 15

As extensions of the results of the Clausen hypergeometric series, we easily show the following results.  q p Property 2.1. If an , cn , z ∈ C, n, p, q ∈ N0 , Re k=1 ck − k=1 ak > 0, and |z| < 1, then the Clausen hypergeometric series of the form ((a) , (c) ; λz)   = p Fq a1 , · · · , ap ; c1 , · · · , cq ; λz   a , · · · , ap ; λz = p Fq 1 c1 , · · · , cq   ∞  (a1 )n · · · ap (λz)n  n = (c1 )n · · · cq n n!

p Fq

(2.5)

n=0

is a solution of the differential equation (Q (q, μ, cn ) ϕ) (z) − (P (p, μ, an , λ) ϕ) (z) = 0,

(2.6)

(Q (q, μ, cn ) ϕ) (z)  q  d  dϕ (z) z =z + (cn − 1) ϕ (z) dz dz n=1 q   

d d  z + (cn − 1) ϕ (z) =z dz dz

(2.7)

(P (p, μ, an , λ) ϕ) (z)  p   dϕ (z) z + an ϕ (z) = λz dz n=1   p   d z + an ϕ (z) . = λz dz

(2.8)

where

n=1

and

n=1

Property 2.2. If an , cn , μ, λ, z ∈ C, n, p, q ∈ N0 , Re the Clausen hypergeometric series of the form

q

k=1 ck

−

p

μp Fq ((a) , (c) ; λz)   = μp Fq a1 , · · · , ap ; c1 , · · · , cq ; λz   a , · · · , ap = μ p Fq 1 ; λz c1 , · · · , cq   ∞  (a1 )n · · · ap (λz)n  n =μ (c1 )n · · · cq n n! n=0

k=1 ak



> 0, and |z| < 1, then

(2.9)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

is a solution of the differential equation (Q (q, μ, λ, cn ) ϕ) (z) − (P (p, μ, λ, an ) ϕ) (z) = 0,

(2.10)

(Q (q, μ, λ, cn ) ϕ) (z)  q  d  dϕ (z) = μq z z + (cn − 1) ϕ (z) dz dz n=1 q  

 d q d z + (cn − 1) ϕ (z) =μ z dz dz

(2.11)

(P (p, μ, λ, an ) ϕ) (z)  p   dϕ (z) p z + an ϕ (z) = μ λz dz n=1   p   d z + an ϕ (z) . = μp λz dz

(2.12)

where

n=1

and

n=1

We now show the derivatives of the Clausen hypergeometric series. Theorem 2.3. The derivative of the Clausen hypergeometric series is given as follows (see [83], p. 315):   d  p Fq a 1 , · · · , ap ; c1 , · · · , cq ; z dz p       an = n=1 p Fq (a1 + 1) , · · · , ap + 1 ; (c1 + 1) , · · · , cq + 1 ; z q n=1 cn

(2.13)

  1 d  p Fq a1 , · · · , ap ; c1 , · · · , cq ; λz λ dz p       an = n=1 p Fq (a1 + 1) , · · · , ap + 1 ; (c1 + 1) , · · · , cq + 1 ; λz , q n=1 cn

(2.14)

and

where an , cn , λ, z ∈ C, n, p, q ∈ N0 , Re

q

k=1 ck

−

p

k=1 ak



> 0, and |z| < 1.

The result was discussed by, for example, Srivastava and Kashyap (1982) [51], Rainville (1960) [45], Andrews et al. (1999) [25] and, Luke (1969) [14]. Without proof, we present extended results for the derivatives of the Clausen hypergeometric series.

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 17

Property 2.3 (Derivatives of the Clausen hypergeometric series). The derivatives for the Clausen hypergeometric series are given as follows [14]:   dn  p Fq a 1 , · · · , ap ; c1 , · · · , cq ; z n dz q   ap        n p Fq (a1 + n) , · · · , ap + n ; (c1 + n) , · · · , cq + n ; z = n=1 q n=1 aq n

(2.15)

and   dn  δ z p Fq a 1 , · · · , ap ; c1 , · · · , cq ; z n dz = (δ + 1 − n)n zδ−n       × p+1 Fq+1 δ + 1, (a1 + 1) , · · · , ap + 1 ; δ + 1 − n, (c1 + 1) , · · · , cq + 1 ; z ,  q p where an , cn , λ, z, δ ∈ C, n, p, q ∈ N0 , Re k=1 ck − k=1 ak > 0, and |z| < 1.

(2.16)

The result was also discussed by, for example, Srivastava and Kashyap (1982) [51], Rainville (1960) [45], and Andrews et al. (1999) [25]. Recalling that δ + 1 − n is a negative integer or zero, we have:   dn  δ z p Fq a 1 , · · · , ap ; c1 , · · · , cq ; z dzn   n! ap =   n−δ aq n−δ (n − δ)! × p+1 Fq+1 ((a1 + n − δ) , · · · , n + 1; n − δ + 1, (c1 + n − δ) , · · · ; z),   d n δ+n−1 z p+1 Fq δ, a1 , · · · , ap ; c1 , · · · , cq ; z n dz     = (δ)n zδ−1 × p+1 Fq (a1 + n) , · · · , ap + n , n + 1; c1 , · · · , cq ; z ,   d n δ−1 z a F , · · · , a ; δ, c , · · · , c ; z p q+1 1 p 1 q dzn     = (δ − n)n zδ−n−1 × p Fq+1 a1 , · · · , ap ; δ − n, (c1 + n) , · · · , cq + n ; z ,

dn δ z (1 − z)a+b−c 2 F1 (a, b; c; z) n dz = (δ − n + 1)n zδ−n × 3 F2 (c − a, c − b, δ + 1, c, δ − n + 1; z),

d n n+c−1 n+a+b−c z F − z) + n, b + n; c + n; z) (1 (a 2 1 dzn = (c)n zc−1 (1 − z)a+b−c 2 F1 (a, b; c; z),

d n c−a+n−1 z (1 − z)a+b−c 2 F1 (a, b; c; z) n dz = (c − a)n zc−a−1 (1 − z)a+b−c−n × 2 F1 (a − n, b; c; z),

(2.17)

(2.18)

(2.19)

(2.20)

(2.21)

(2.22)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

d n c−1 a+b−c z F − z) b; c; z) (1 (a, 2 1 dzn = (c − n)n zc−n−1 (1 − z)a+b−c−n × 2 F1 (a − n, b − n; c − n; z),

dn a+b−c F b; c; z) − z) (a, (1 2 1 dzn (c − n)n (c − b)n = (1 − z)a+b−c−n × 2 F1 (a, b; c + n; z), (c)n

dn a b z − z) (1 dzn = (a − n + 1)n za−n (1 − z)b−n × 2 F1 (−n, a + b + 1; a + 1 − n; z) (a < n) ,

dn a b z − z) (1 dzn n! (−b)n−a = (1 − z)b−n (n − a)! × 2 F1 (−a, b + 1; n − a + 1; z) (a < n) ,

d n c−1 b−c+n z F − z) b; c; z) (1 (a, 2 1 dzn c−1−n = (c − n)n z (1 − z)b−c

(2.23)

(2.24)

(2.25)

(2.26)

(2.27)

× 2 F1 (a − n, b; c − n; z), and

dn a+n−1 F b; c; z) − z) (a, (1 1 2 dzn (−1)n (a)n (c − b)n = (1 − z)a−1 (c)n

(2.28)

× 2 F1 (a − n, b; c + n; z) , q  p where a1 , · · · , ap ; c1 , · · · , cq , δ, a, b, c ∈ C, Re k=1 ck − k=1 ak > 0, and n ∈ N0 . Returning now to the details of the derivatives of the Clausen hypergeometric series, see Srivastava and Kashyap (1982) [51], Rainville (1960) [45], and Andrews et al. (1999) [25], Luke (1969) [14], Andrews (1992) [87], and Erdilyi et al. (1953) [88]. Theorem 2.4. [Rainville (1960)] q     If p ≤ q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, Re k=1 ck −  p k=1 ak > 0, and |z| < 1, then (see [45], p. 85)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 19  p Fq

a1 , · · · , ap ; λz c 1 , · · · , cq ;



 (c1 ) =  (a1 )  (c1 − a1 )



1

t

a1 −1

(1 − t)

c1 −a1 −1

  p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt,

(2.29)

0

where λ is a constant. When c1 − a1 = 1, we have the following result.

    Corollary 2.1. If p ≤ q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c2 ) > 0, · · · , Re cq > 0, q p Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

 a 1 , · · · , ap ; λz p Fq a1 + 1, · · · , cq ;     (a1 + 1) 1 a1 −1 t = p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt,  (a1 ) 0

(2.30)

where λ is a constant. When a1 = 1, we know the following result.

    Corollary 2.2. If p ≤ q + 1, Re (a2 ) > 0, · · · , Re ap > 0, Re (c1 ) > 1, · · · , Re cq > 0, q p Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

 1, · · · , ap ; λz p Fq c 1 , · · · , cq ;  1    (c1 ) = (1 − t)c1 −2 p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt,  (c1 − 1) 0

(2.31)

where λ is a constant. Theorem 2.5. [Askey (1975)] q     If p ≤  q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, Re k=1 ck − p k=1 ak > 0, and |z| < 1, then we have (see [60], p. 19)   a1 , · · · , ap , a; λz c1 , · · · , cq , c;  1    (c) t a−1 (1 − t)c−a−1 p Fq a1 , a2 , · · · , ap ; c1 , c2 , · · · , cq ; λzt dt, =  (a)  (c − a) 0

p+1 Fq+1

(2.32)

where λ is a constant. The results were obtained by Rainville in 1960 (see [45], p. 85) and Askey (1975) [60], and further investigated by Andrews et al. (see [25], p. 67), Srivastava and Kashyap (1982) [51], and Slater (1966) [39]. When c − a = 1, we get the following result.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

    Corollary 2.3. If p ≤ q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, q  p Re k=1 ck − k=1 ak > 0, and |z| < 1, then we have 

 a1 , · · · , ap , a; λz p+1 Fq+1 c1 , · · · , cq , a + 1;     (a + 1) 1 a−1 t = p Fq a1 , a2 , · · · , ap ; c1 , c2 , · · · , cq ; λzt dt,  (a) 0

(2.33)

where λ is a constant. When a = 1, we get the following result.   Corollary 2.4. If p ≤ q + 1, Re (a1 ) > 0, · · · , Re (c) > 1, Re ap > 0, Re (c1 ) > 0, · · · , q    p Re cq > 0, Re k=1 ck − k=1 ak > 0, and |z| < 1, then we have   a1 , · · · , ap , 1; λz p+1 Fq+1 c1 , · · · , cq , c;  1    (c) = (1 − t)c−2 p Fq a1 , a2 , · · · , ap ; c1 , c2 , · · · , cq ; λzt dt,  (c − 1) 0

(2.34)

where λ is a constant. Theorem 2.6. [Rainville (1960)]     If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, |z| < 1, q  p Re k=1 ck − k=1 ak > 0, and κ ∈ N, then 

  t α−1 (z − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt 0

α α+1 α+κ−1 , · · · , a , , , · · · , ; a 1 p κ κ κ κ = zα+β−1 × p+κ Fq+κ , α+β+1 α+β+κ−1 λz , , · · · , ; c1 , · · · , cq , α+β κ κ κ

1 B (α, β)

z

(2.35)

where λ is a constant. The result was obtained by Rainville in 1960 (see [45], p. 85). Based on this result, we suggest the following corollaries.     Corollary 2.5. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, |z| < 1, q  p Re k=1 ck − k=1 ak > 0, and κ ∈ N, then

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 21 

  t β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)κ dt 0

a1 , · · · , ap , κ1 , κ2 , · · · , 1; β κ = z × p+κ Fq+κ , β+2 β+κ λz c1 , · · · , cq , 1+β κ , κ ,··· , κ ;

1 B (1, β)

z

(2.36)

where λ is a constant.     Corollary 2.6. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, |z| < 1, q  p Re k=1 ck − k=1 ak > 0, and κ ∈ N, then 

  t β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t) dt 0   a1 , · · · , ap , 1; λz , = zβ × p+1 Fq+1 c1 , · · · , cq , 1 + β;

1 B (1, β)

z

(2.37)

where λ is a constant. Using the above relations, we have the following particular cases.   Case 1. Suppose that Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , q    p Re cq > 0, Re k=1 ck − k=1 ak > 0, and |z| < 1. Then we have 



t α−1 (z − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt 2 dt 0

a1 , · · · , ap , α2 , α+1 2 ; α+β−1 2 =z × p+2 Fq+2 , α+β+1 λz ; c1 , · · · , cq , α+β 2 , 2

1 B (α, β)

z

(2.38)

where λ is a constant.   Case 2. Suppose that Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , q    p Re cq > 0, Re k=1 ck − k=1 ak > 0, and |z| < 1. Then we have 



t α−1 (z − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt 3 dt 0

α+2 a1 , · · · , ap , α3 , α+1 3 , 3 ; α+β−1 3 =z × p+3 Fq+3 , α+β+1 α+β+2 λz c1 , · · · , cq , α+β , 3 ; 3 , 3

1 B (α, β)

z

(2.39)

where λ is a constant.   Case 3. Suppose that Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , q    p Re cq > 0, Re k=1 ck − k=1 ak > 0, and |z| < 1. Then we have

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 



t α−1 (z − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt 4 dt 0

α+2 α+3 a1 , · · · , ap , α4 , α+1 4 , 4 , 4 ; α+β−1 4 =z × p+4 Fq+4 , α+β+1 α+β+2 α+β+3 λz , 4 , 4 ; c1 , · · · , cq , α+β 4 , 4

1 B (α, β)

z

(2.40)

where λ is a constant. Case 4. Suppose that Re (α) > 0, Re (β) > 0, and Re (β) > Re (α) > 0. Then we have

 z α α+1 α+2 1 3, 3 , 3 ; α−1 β−1 λt 3 α+β−1 3 t e dt = z × 3 F3 α+β α+β+1 α+β+2 λz , (z − t) B (α, β) 0 , 3 ; 3 , 3

(2.41)

where λ is a constant. Case 5. Suppose that Re (α) > 0, Re (β) > 0, |z| < 1, Re (β) > Re (α) > 0, and κ ∈ N, then we have  z 1 κ t α−1 (z − t)β−1 eλt dt B (α, β) 0

(2.42) α α+1 α+κ−1 ; κ , κ ,··· , κ α+β−1 κ =z λz F , κ κ α+β α+β+1 , · · · , α+β+κ−1 ; κ , κ κ where λ is a constant. Theorem 2.7. [Rainville (1960)]     > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, (α) > 0, Re (β) IfRe   q p Re k=1 ck − k=1 ak > 0, and |z| < 1, then  z   1 t α−1 (z − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt dt B (α, β) 0   (2.43) a1 , · · · , ap , α; λz , = zα+β−1 × p+1 Fq+1 c1 , · · · , cq , α + β; where λ is a constant. The result was obtained by Rainville in 1960 [45]. Theorem 2.8. [Manocha and Srivastava (1984)]    (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and IfRe  q p Re k=1 ck − k=1 ak > 0, then 

  t α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; λ , = p+1 Fq+1 c1 , · · · , cq , α + β;

1 B (α, β)

where λ is a constant.

1

(2.44)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 23

The result was obtained by Manocha and Srivastava in 1984 [22]. Using this formula, we obtain the following results. Theorem 2.9. If Re (α) > 0, Re (β) > 0, and |z| < 1, then 1 B (α, β)





z

t

α−1

(z − t)

0

e dt = z

β−1 λt

α+β−1

× 1 F1

 α; λz , α + β;

(2.45)

where λ is a constant. Proof. Putting eλt = 0 F0 (−; −; λz), we simply write  z 1 t α−1 (z − t)β−1 eλt dt B (α, β) 0  z 1 t α−1 (z − t)β−1 0 F0 (−; −; λt)dt. = B (α, β) 0

(2.46)

It is easy to verify that 1 B (α, β)



z

t α−1 (z − t)β−1 0 F0 (−; −; λt)dt   α; = zα+β−1 × 1 F1 λz . α + β; 0

(2.47)

Thus we obtain the result. Similarly, without proofs, we may present the following theorems. Theorem 2.10. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c1 ) > Re (a1 ), and |z| < 1, then  z 1 t α−1 (z − t)β−1 1 F1 (a1 ; c1 ; λt) dt B (α, β) 0   (2.48) a1 , α; α+β−1 λz , × 2 F2 =z c1 , α + β; where λ is a constant. Theorem 2.11. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, and |z| < 1, then  z 1 t α−1 (z − t)β−1 2 F2 (a1 , a2 ; c1 , c2 ; λt) dt B (α, β) 0   (2.49) a1 , a2 , α; λz , = zα+β−1 × 3 F3 c1 , c2 , α + β; where λ is a constant.

24

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 2.12. If Re (α) > 0, Re (β) > 0, Re (c) > 0, and |z| < 1, then    z 1 α; t α−1 (z − t)β−1 0 F1 (−; c; λt) dt = zα+β−1 × 1 F2 λz , c, α + β; B (α, β) 0

(2.50)

where λ is a constant. Theorem 2.13. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, |z| < 1, and κ ∈ N, then  z 1 t α−1 (z − t)β−1 1 F2 (a1 ; c1 , c2 ; λt) dt B (α, β) 0   (2.51) a1 , α; α+β−1 λz , =z 2 F3 c1 , c2 , α + β; where λ is a constant. Theorem 2.14. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then    z 1 a1 , α; α−1 β−1 α+β−1 t λz , (z − t) 1 F0 (a1 ; −; λt) dt = z 2 F1 α + β; B (α, β) 0

(2.52)

where λ is a constant. Theorem 2.15. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then    z 1 a1 , α; α−1 β−1 −a1 α+β−1 t dt = z z . (2.53) (z − t) (1 − t) 2 F1 α + β; B (α, β) 0   Theorem 2.16. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0 and |z| < 1, then  z

1 t α−1 (z − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt 2 dt B (α, β) 0

(2.54) a1 , · · · , ap , α2 , α+1 2 ; α+β−1 2 =z λz , p+2 Fq+2 α+β+1 ; c1 , · · · , cq , α+β 2 , 2 where λ is a constant. Theorem 2.17. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, and |z| < 1, then  z

1 t α−1 (z − t)β−1 1 F1 a1 ; c1 ; λt 2 dt B (α, β) 0

α α+1 , , ; a 1 2 2 2 = zα+β−1 3 F3 , α+β+1 λz vc1 , α+β ; 2 , 2 where λ is a constant.

(2.55)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 25

Theorem 2.18. If Re (α) > 0, Re (β) > 0, and |z| < 1, then 1 B (α, β)



z

t α−1 (z − t)β−1 e

λt 2

0

dt = zα+β−1 × 2 F2

α α+1 2, 2 ; 2 α+β α+β+1 λz , ; 2 2

,

(2.56)

where λ is a constant. Theorem 2.19. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then  z

1 t α−1 (z − t)β−1 1 F0 a1 ; −; λt 2 dt B (α, β) 0

a1 , α2 , α+1 2 ; α+β−1 2 =z , 3 F2 α+β α+β+1 λz ; 2 , 2

(2.57)

where λ is a constant. Theorem 2.20. If Re (α) > 0, Re (β) > 0, Re (c1 ) > 0, and |z| < 1, then  z

1 t α−1 (z − t)β−1 0 F1 −; c1 ; λt 2 dt B (α, β) 0

α α+1 2, 2 ; α+β−1 2 =z , 2 F3 α+β+1 λz c1 , α+β ; 2 , 2

(2.58)

where λ is a constant. Theorem 2.21. If Re (α) > 0, Re (β) > 0, |z| < 1, and κ ∈ N, then  z 1 κ t α−1 (z − t)β−1 eλt dt B (α, β) 0

α α+1 α+κ−1 ; κ , κ ,··· , κ α+β−1 κ =z λz , κ Fκ α+β α+β+1 , · · · , α+β+κ−1 ; κ , κ κ

(2.59)

where λ is a constant and κ ∈ N.     Theorem 2.22. Suppose that Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, Re (α) > 0, and Re (β) > 0. Then  0

=

2

  t α−1 (2 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt dt



1

−1

(1 + t)

α−1

(1 − t)

β−1

  p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t) dt,

where λ is a constant. Proof. On changing the variable to 2 − t = 1 − τ , we obtain the result.

(2.60)

26

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

    Theorem 2.23. If Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, Re (α) > 0, and Re (β) > 0, then 1 B (α, β) =2



1 −1

α+β−1

  (1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t) dt 

× p+1 Fq+1

 a1 , · · · , ap , α; 2λ , c1 , · · · , cq , α + β;

(2.61)

where λ is a constant. Proof. Using the equality 1 B (α, β)



2

  t α−1 (2 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt dt

0

= 2α+β−1



a1 , · · · , ap , α; c1 , · · · , cq , α + β;

× p+1 Fq+1

(2.62)

 2λ ,

we have 

  (1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t) dt −1   = 2α+β−1 × p+1 Fq+1 a1 , · · · , ap , α;c1 , · · · , cq , α + β; 2λ

1 B (α, β)

1

(2.63)

in association with  2   t α−1 (2 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0

=



1

−1

  (1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t) dt.

(2.64)

Theorem 2.24. If Re (α) > 0 and Re (β) > 0, then 1 B (α, β)



1 −1

 (1 + t)α−1 (1 − t)β−1 eλ(1+t) dt = 2α+β−1 × 1 F1

 α; 2λ , α + β;

(2.65)

where λ is a constant. Proof. In view of the integral representation 1 B (α, β)





2

t 0

α−1

(2 − t)

e dt = 2

β−1 λt

α+β−1

1 F1

 α; 2λ , α + β;

(2.66)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 27

we can write 1 B (α, β) since



2



1

−1

 (1 + t)α−1 (1 − t)β−1 eλ(1+t) dt = 2α+β−1 1 F1

 t α−1 (2 − t)β−1 eλt dt =

0

1

−1

α; 2λ α + β;

(1 + t)α−1 (1 − t)β−1 eλ(1+t) dt.

 (2.67)

(2.68)

Theorem 2.25. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then 1 B (α, β)



1 −1

(1 + t)α−1 (1 − t)β−1 1 F1 (a1 ; c1 ; λ (1 + t)) dt

= 2α+β−1   a1 , α; 2λ , × 2 F2 c1 , α + β;

(2.69)

where λ is a constant. Proof. By the equality 1 B (α, β)



2

t α−1 (2 − t)β−1 1 F1 (a1 ; c1 ; λt) dt   a1 , α; α+β−1 2λ =2 2 F2 c1 , α + β; 0

(2.70)

we have 1 B (α, β) =2



1 −1

α+β−1

(1 + t)α−1 (1 − t)β−1 1 F1 (a1 ; c1 ; λ (1 + t)) dt

2 F2



 a1 , α; 2λ , c1 , α + β;

(2.71)

from which it follows that  0

=

2

t α−1 (2 − t)β−1 1 F1 (a1 ; c1 ; λt) dt



1

−1

(2.72) (1 + t)

α−1

(1 − t)

β−1

1 F1 (a1 ; c1 ; λ (1 + t)) dt.

28

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 2.26. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and Re (c2 ) > 0, then  1 1 (1 + t)α−1 (1 − t)β−1 2 F2 (a1 , a2 ; c1 , c2 ; λ (1 + t)) dt B (α, β) −1 = 2α+β−1   a1 , a2 , α; 2λ , × 3 F3 c1 , c2 , α + β;

(2.73)

where λ is a constant. Proof. Using the equality 1 B (α, β)



2

t α−1 (2 − t)β−1 2 F2 (a1 , a2 ; c1 , c2 ; λt) dt

0

= 2α+β−1   a1 , a2 , α; 2λ , × 3 F3 c1 , c2 , α + β;

(2.74)

we get 1 B (α, β) =2



1

−1

α+β−1

(1 + t)α−1 (1 − t)β−1 2 F2 (a1 , a2 ; c1 , c2 ; λ (1 + t)) dt 

× 3 F3

 a1 , a2 , α; 2λ , c1 , c2 , α + β;

(2.75)

where 

2

0

t α−1 (2 − t)β−1 2 F2 (a1 , a2 ; c1 , c2 ; λt) dt



=

1

−1

(2.76) (1 + t)

α−1

(1 − t)

β−1

2 F2 (a1 , a2 ; c1 , c2 ; λ (1 + t)) dt.

Theorem 2.27. If Re (α) > 0, Re (β) > 0, and Re (c1 ) > 0, then 1 B (α, β) =z where λ is a constant.



α+β−1

1

−1

(1 + t)α−1 (1 − t)β−1 0 F1 (−; c; λ (1 + t)) dt

1 F2



 α; 2λ , c, α + β;

(2.77)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 29

Proof. By the expression 1 B (α, β)



2

t α−1 (2 − t)β−1 0 F1 (−; c; λt) dt 0   α; α+β−1 2λ F =z 1 2 c, α + β;

(2.78)

we have 1 B (α, β) =z



1

−1

α+β−1

(1 + t)α−1 (1 − t)β−1 0 F1 (−; c; λ (1 + t)) dt 

1 F2

 α; 2λ , c, α + β;

from which it follows that  2 1 t α−1 (2 − t)β−1 0 F1 (−; c; λt) dt B (α, β) 0  1 1 = (1 + t)α−1 (1 − t)β−1 0 F1 (−; c; λ (1 + t)) dt. B (α, β) −1

(2.79)

(2.80)

Theorem 2.28. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, and Re (c2 ) > 0, then 1 B (α, β) =2



1 −1

α+β−1

(1 + t)α−1 (1 − t)β−1 1 F2 (a1 ; c1 , c2 ; λ (1 + t)) dt 

2 F3

 a1 , α; 2λ , c1 , c2 , α + β;

(2.81)

where λ is a constant. Proof. By the equality 1 B (α, β)



2

t α−1 (2 − t)β−1 1 F2 (a1 ; c1 , c2 ; λt) dt 0   a1 , α; α+β−1 2λ =2 2 F3 c1 , c2 , α + β;

(2.82)

we have 1 B (α, β) =2



α+β−1

1 −1

(1 + t)α−1 (1 − t)β−1 1 F2 (a1 ; c1 , c2 ; λ (1 + t)) dt

2 F3



 a1 , α; 2λ , c1 , c2 , α + β;

(2.83)

30

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

in which 

2

0

t α−1 (2 − t)β−1 1 F2 (a1 ; c1 , c2 ; λt) dt



=

1

−1

(2.84) (1 + t)

α−1

(1 − t)

β−1

1 F2 (a1 ; c1 , c2 ; λ (1 + t)) dt.

Theorem 2.29. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then 1 B (α, β) =2



1 −1

α+β−1

(1 + t)α−1 (1 − t)β−1 1 F0 (a1 ; −; λ (1 + t)) dt 

2 F1

 a1 , α; 2λ , α + β;

(2.85)

where λ is a constant. Proof. In consequence of 1 B (α, β)



2

t α−1 (2 − t)β−1 1 F0 (a1 ; −; λt) dt   a1 , α; α+β−1 =2 2λ , 2 F1 α + β; 0

(2.86)

we can write 

1 B (α, β)

1

−1

(1 + t)α−1 (1 − t)β−1 1 F0 (a1 ; −; λ (1 + t))dt 

= 2α+β−1 2 F1

 a1 , α; 2λ , α + β;

(2.87)

since 

2

0

t α−1 (2 − t)β−1 1 F0 (a1 ; −; λt) dt



=

1

−1

(2.88) (1 + t)

α−1

(1 − t)

β−1

1 F0 (a1 ; −; λ (1 + t))dt.

Theorem 2.30. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then 1 B (α, β)



1

−1

 (1 + t)

α−1

(1 − t)

β−α−1

dt = 2

α+β−1

2 F1

 a1 , α; 2 . α + β;

(2.89)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 31

Proof. Using 1 B (α, β)



2

t α−1 (2 − t)β−1 (1 − t)−a dt = 2α+β−1 2 F1

0



 a1 , α; 2 , α + β;

(2.90)

we derive 1 B (α, β)



1

−1

1 = B (α, β)

(1 + t)α−1 (1 − t)β−1 (1 − t)−a dt



1

(1 + t)α−1 (1 − t)β−α−1 dt   a1 , α; α+β−1 =2 2 , 2 F1 α + β; −1

which follows from  2  t α−1 (2 − t)β−1 (1 − t)−a dt =

1

−1

0

(1 + t)α−1 (1 − t)β−1 (2 − t)−a dt.

(2.91)

(2.92)

    Theorem 2.31. If Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, Re (α) > 0, and Re (β) > 0, then 



(1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)2 dt −1

a1 , · · · , ap , α2 , α+1 2 ; α+β−1 =2 4λ , p+2 Fq+2 α+β+1 c1 , · · · , cq , α+β ; 2 , 2

1 B (α, β)

1

(2.93)

where λ is a constant. Proof. Using the equality 



t α−1 (2 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt 2 dt 0

a1 , · · · , ap , α2 , α+1 2 ; α+β−1 =2 p+2 Fq+2 α+β+1 4λ , c1 , · · · , cq , α+β ; 2 , 2

1 B (α, β)

2

(2.94)

we arrive at the following result: 



(1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)2 dt −1

a1 , · · · , ap , α2 , α+1 2 ; α+β−1 =2 4λ , p+2 Fq+2 α+β+1 ; c1 , · · · , cq , α+β 2 , 2

1 B (α, β)

1

(2.95)

32

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

in which



2

0



t α−1 (2 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt 2 dt



=

1

−1

(2.96)



(1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)2 dt.

Theorem 2.32. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then  1

1 (1 + t)α−1 (1 − t)β−1 1 F1 a1 ; c1 ; λ (1 + t)2 dt B (α, β) −1

a1 , α2 , α+1 2 ; α+β−1 =2 3 F3 α+β+1 4λ , c1 , α+β ; 2 , 2

(2.97)

where λ is a constant. Proof. By the equality 



t α−1 (2 − t)β−1 1 F1 a1 ; c1 ; λt 2 dt 0

a1 , α2 , α+1 2 ; α+β−1 =2 3 F3 α+β+1 4λ c1 , α+β ; 2 , 2

1 B (α, β)

we write



2

(2.98)



(1 + t)α−1 (1 − t)β−1 1 F1 a1 ; c1 ; λ (1 + t)2 dt −1

a1 , α2 , α+1 2 ; α+β−1 =2 4λ , 3 F3 α+β+1 ; c1 , α+β 2 , 2

1 B (α, β)

1

(2.99)

since we can easily verify that  2

t α−1 (2 − t)β−1 1 F1 a1 ; c1 ; λt 2 dt 0

=



1 −1

(1 + t)

α−1

(1 − t)

β−1

1 F1

a1 ; c1 ; λ (1 + t)

2

(2.100)

dt.

Theorem 2.33. If Re (α) > 0 and Re (β) > 0, then 1 B (α, β)



1

−1

(1 + t)

where λ is a constant.

α−1

(1 − t)

β−1 λ(1+t)2

e

dt = 2

α+β−1

× 2 F2



α α+1 2, 2 ; α+β α+β+1 4λ ; 2 , 2

,

(2.101)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 33

Proof. Using the equation 1 B (α, β)





2

t

α−1

β−1 λt 2

(2 − t)

e

dt = 2

α+β−1

× 2 F2

0

α α+1 2, 2 ; α+β α+β+1 4λ ; 2 , 2

,

we can conclude that

α α+1  1 1 2, 2 ; α−1 β−1 λ(1+t)2 α+β−1 e dt = 2 × 2 F2 α+β α+β+1 4λ , (1 + t) (1 − t) B (α, β) −1 ; 2 , 2 since



2

2



t α−1 (2 − t)β−1 eλt dt =

1

−1

0

2

(1 + t)α−1 (1 − t)β−1 eλ(1+t) dt.

Theorem 2.34. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then  1

1 (1 + t)α−1 (1 − t)β−1 1 F0 a1 ; −; λ (1 + t)2 dt B (α, β) −1

a1 , α2 , α+1 2 ; α+β−1 =2 3 F2 α+β α+β+1 4λ , ; 2 , 2

(2.102)

(2.103)

(2.104)

(2.105)

where λ is a constant. Proof. Using the equation





t α−1 (2 − t)β−1 1 F0 a1 ; −; λt 2 dt 0

a1 , α2 , α+1 2 ; α+β−1 =2 3 F2 α+β α+β+1 4λ , ; 2 , 2

1 B (α, β)

we get



2

(2.106)



(1 + t)α−1 (1 − t)β−1 1 F0 a1 ; −; λ (1 + t)2 dt −1

a1 , α2 , α+1 2 ; α+β−1 =2 3 F2 α+β α+β+1 4λ , ; 2 , 2

1 B (α, β)

1

(2.107)

since we can easily verify that  2

t α−1 (2 − t)β−1 1 F0 a1 ; −; λt 2 dt 0

=



1 −1

(1 + t)

α−1

(1 − t)

β−1

1 F0

a1 ; −; λ (1 + t)

2

(2.108)

dt.

34

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 2.35. If Re (c1 ) > 0, Re (α) > 0, and Re (β) > 0, then 



(1 + t)α−1 (1 − t)β−1 0 F1 −; c1 ; λ (1 + t)2 dt −1

α α+1 2, 2 ; α+β−1 =z 2 F3 α+β+1 4λ , ; c1 , α+β 2 , 2

1 B (α, β)

1

(2.109)

where λ is a constant. Proof. By the equality 



t α−1 (2 − t)β−1 0 F1 −; c1 ; λt 2 dt 0

α α+1 2, 2 ; α+β−1 =z 4λ 2 F3 α+β+1 c1 , α+β ; 2 , 2

1 B (α, β)

2

(2.110)

we find that 

1 B (α, β)

1 −1

1 = B (α, β)



(1 + t)α−1 (1 − t)β−1 0 F1 −; c1 ; λ (1 + t)2 dt



1

−1

(1 + t)

α−1

(1 − t)

β−1



2 0 F1 −; c1 ; λ (1 + t) dt,

(2.111)

since we can easily verify that 

2

0

=



t α−1 (2 − t)β−1 0 F1 −; c1 ; λt 2 dt



1 −1

(1 + t)

α−1

(1 − t)

β−1

0 F1

(2.112)

−; c1 ; λ (1 + t) dt. 2

Theorem 2.36. If Re (α) > 0, Re (β) > 0, and κ ∈ N, then 1 B (α, β)



1

κ

(1 + t)α−1 (1 − t)β−1 eλ(1+t) dt −1

α α+1 α+κ−1 , , · · · , ; κ κ κ = 2α+β−1 κ Fκ α+β κ α+β+1 , α+β+κ−1 λ2 , , · · · , ; κ κ κ

where λ is a constant.

(2.113)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 35

Proof. In view of 1 B (α, β)



2

κ

t α−1 (2 − t)β−1 eλt dt 0

α α+1 α+κ−1 ; κ , κ ,··· , κ α+β−1 κ =2 λ2 , κ Fκ α+β α+β+1 , · · · , α+β+κ−1 ; κ , κ κ

we find that 1 B (α, β)



1

(2.114)

κ

(1 + t)α−1 (1 − t)β−1 eλ(1+t) dt −1

α α+1 α+κ−1 ; κ , κ ,··· , κ α+β−1 κ =2 λ2 , κ Fκ α+β α+β+1 , · · · , α+β+κ−1 ; κ , κ κ

from which it follows that   2 κ t α−1 (2 − t)β−1 eλt dt = 0

1

−1

(2.115)

κ

(1 + t)α−1 (1 − t)β−1 eλ(1+t) dt.

More generally, we easily see that

 3 a, β; 9 α−1 β−1 t α+β−1 t e dt = B (α, β) 3 , (3 − t) 2 F2 α+β α+β+1 ;4 0 2 , 2

 4 a, β; α−1 β−1 t α+β−1 t e dt = B (α, β) 4 (4 − t) 2 F2 α+β α+β+1 4 , ; 0 2 , 2

 5 a, β; 25 α−1 β−1 t α+β−1 t e dt = B (α, β) 5 , (z − t) 2 F2 α+β α+β+1 ; 4 0 2 , 2

 6 a, β; 36 α−1 β−1 t α+β−1 t e dt = B (α, β) 6 , (6 − t) 2 F2 α+β α+β+1 ; 4 0 2 , 2

 7 a, β; 49 α−1 β−1 t α+β−1 t e dt = B (α, β) 7 , (7 − t) 2 F2 α+β α+β+1 ; 4 0 2 , 2

 8 a, β; α−1 β−1 t α+β−1 t e dt = B (α, β) 8 (8 − t) 2 F2 α+β α+β+1 16 , ; 0 2 , 2

 9 a, β; 81 α−1 β−1 t α+β−1 t e dt = B (α, β) 9 , (9 − t) 2 F2 α+β α+β+1 ; 4 0 2 , 2

 10 a, β; α−1 β−1 t α+β−1 t e dt = B (α, β) 10 (10 − t) 2 F2 α+β α+β+1 25 , ; 0 2 , 2

(2.116)

(2.117)

(2.118)

(2.119)

(2.120)

(2.121)

(2.122)

(2.123)

(2.124)

36

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and .. . 



n

t

α−1

(n − t)

e dt = B (α, β) n

β−1 t

α+β−1

2 F2

0

n2 α+β α+β+1 4 , ; 2 , 2 a, β;

(2.125)

where Re (α) > 0 and Re (β) > 0. Theorem 2.37. [Rainville (1960)]     If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have  z   t α−1 (z − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt 0

= B (α, β) zα+β−1

, βs , · · · , β+s−1 ; κ κ s s λzκ+s a1 , · · · , ap , ακ , · · · , α+κ−1 κ s × p+κ+s Fq+κ+s κ+s , α+β+κ+s−1 + s) (κ , · · · , ; c1 , · · · , cq , α+β κ+s κ+s

(2.126)

where λ is a constant. The result was obtained by Rainville in 1960 (see [45], p. 104).   Theorem 2.38. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have 1 B (α, β)

 0

1

  t α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt

, βs , · · · , β+s−1 ; κ κ ss λ a1 , · · · , ap , ακ , · · · , α+κ−1 κ s

= p+κ+s Fq+κ+s

α+β+κ+s−1 c1 , · · · , cq , α+β ; κ+s , · · · , κ+s

(κ + s)κ+s

(2.127) ,

where λ is a constant. Proof. Recalling the formula and putting z = 1, we conclude the result.   Theorem 2.39. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have 1 B (α, β)



1

−1

  (1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt

= 2α+β−1



, βs , · · · , β+s−1 ; κ κ s s λ2κ+s a1 , · · · , ap , ακ , · · · , α+κ−1 κ s × p+κ+s Fq+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c1 , · · · , cq , α+β κ+s , · · · , κ+s where λ is a constant.

(2.128)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 37

Proof. We may get the result by using the formula and putting z = 2. Theorem 2.40. If Re (α) > 0, Re (β) > 0, κ, s ∈ N, and |z| < 1, then we have 

z

t α−1 (z − t)β−1 eλt

κ (z−t)s

dt

0

= B (α, β) zα+β−1 α α+1

α+κ−1 β β+1 , s , s , · · · , β+s−1 ; κ κ s s λzκ+s κ , κ ,··· , κ s × κ+s Fκ+s , α+β α+β+1 α+β+κ+s−1 (κ + s)κ+s ; κ+s , κ+s , · · · , κ+s

(2.129)

where λ is a constant. Proof. We get the result by using the formula eλt

κ (z−t)s

  = 0 F0 λt κ (z − t)s .

Theorem 2.41. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, κ, s ∈ N, and |z| < 1, then we have  z   t α−1 (z − t)β−1 1 F1 a1 ; c1 ; λt κ (z − t)s dt 0

= B (α, β) zα+β−1

, βs , · · · , β+s−1 ; κ κ s s λzκ+s a1 , ακ , · · · , α+κ−1 κ s × 1+κ+s F1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c1 , α+β κ+s , · · · , κ+s

(2.130)

where λ is a constant. Proof. We obtain the result by taking p = 1 and q = 1,. Theorem 2.42. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, κ, s ∈ N, and |z| < 1, then we have 

z

  t α−1 (z − t)β−1 2 F2 a1 , a2 ; c1 , c2 ; λt κ (z − t)s dt

0

= B (α, β) zα+β−1

, βs , · · · , β+s−1 ; κ κ s s λzκ+s a1 , a2 , ακ , · · · , α+κ−1 κ s × 2+κ+s F2+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c1 , c2 , α+β κ+s , · · · , κ+s where λ is a constant. Proof. We obtain the result by taking p = 2 and q = 2.

(2.131)

38

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 2.43. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, κ, s ∈ N, and |z| < 1, then we have 

z

0

t α−1 (z − t)β−1  a dt 1 − λt κ (z − t)s 1

= B (α, β) zα+β−1

, βs , · · · , β+s−1 ; κ κ s s λzκ+s a1 , ακ , · · · , α+κ−1 κ s × 1+κ+s Fκ+s , α+β α+β+κ+s−1 (κ + s)κ+s ; κ+s , · · · , κ+s

(2.132)

where λ is a constant. Proof. Putting p = 1 and q = 0, that is, 1 F0

   −a a1 ; −; λt κ (1 − t)s = 1 − λt κ (z − t)s 1 ,

(2.133)

we get the result. Theorem 2.44. If Re (α) > 0, Re (β) > 0, Re (c1 ) > 0, κ, s ∈ N, and |z| < 1, then we have 

z

  t α−1 (z − t)β−1 0 F1 −; c1 ; λt κ (z − t)s dt

0

= B (α, β) zα+β−1 α

α+κ−1 β , s , · · · , β+s−1 ; κ κ s s λzκ+s κ ,··· , κ s × κ+s F1+κ+s κ+s , α+β+κ+s−1 ; (κ + s) c1 , α+β κ+s , · · · , κ+s

(2.134)

where λ is a constant. Proof. We obtain the result by taking p = 0 and q = 1. Theorem 2.45. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, κ, s ∈ N, and |z| < 1, then we have  z   t α−1 (z − t)β−1 2 F1 a1 , a2 ; c1 ; λt κ (z − t)s dt 0

= B (α, β) zα+β−1

, βs , · · · , β+s−1 ; κ κ s s λzκ+s a1 , a2 , ακ , · · · , α+κ−1 κ s × 2+κ+s F1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c1 , α+β κ+s , · · · , κ+s where λ is a constant. Proof. Taking p = 2 and q = 1, we get the result.

(2.135)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 39

Theorem 2.46. If Re (α) > 0, Re (β) > 0, and |z| < 1, then we have  z t α−1 (z − t)β−1 eλt(z−t) dt 0

2 α, β; λz = B (α, β) zα+β−1 2 F2 α+β α+β+1 , ; 4 2 , 2

(2.136)

where λ is a constant. Proof. Taking p = 2 and q = 1, that is, 0 F0

  −; −; λt κ (z − t)s = eλt(z−t) ,

(2.137)

we get the result. Theorem 2.47. If Re (α) > 0, Re (β) > 0, and |z| < 1, then we have  z 2 t α−1 (z − t)β−1 eλt(z−t) dt 0

α, β2 , β+1 4 2 2 ; α+β−1 λz , = B (α, β) z 3 F3 α+β α+β+1 α+β+2 , , ;9 3

3

(2.138)

3

where λ is a constant. Proof. Taking p = 0 and q = 0, that is, 0 F0 (−; −; λt (z − t)) = e

λt(z−t)

,

(2.139)

we obtain the result. Theorem 2.48. If Re (α) > 0, Re (β) > 0, and |z| < 1, then we have  z 2 t α−1 (z − t)β−1 eλt (z−t) dt 0

β α, α+1 4 2 2 , 2; α+β−1 = B (α, β) z λz , 3 F3 α+β α+β+1 α+β+2 , , ;9 3

3

(2.140)

3

where λ is a constant. Proof. Taking p = 0 and q = 0, that is,

2 λt 2 (z−t) , 0 F0 −; −; λt (z − t) = e we get the result.

(2.141)

40

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 2.49. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then we have 

z

0

t α−1 (z − t)β−1  a dt 1 − λt (z − t)2 1

= B (α, β) z

α+β−1

4 F3

4 2 , α+β α+β+1 α+β+2 9 λz , 3 ; 3 , 3 a1 , α, α+1 2 , β;

(2.142)

where λ is a constant. Proof. Putting p = 1 and q = 0, that is, 1 F0



−a1 a1 ; −; λt (1 − t)2 = 1 − λt (z − t)2 ,

(2.143)

we deduce the result. Theorem 2.50. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then we have  0

z

t α−1 (z − t)β−1  a dt 1 − λt 2 (z − t) 1

= B (α, β) z

α+β−1

4 F3

4 2 λz , α+β α+β+1 α+β+2 9 , 3 ; 3 , 3 a1 , α, β, β+1 2 ;

(2.144)

where λ is a constant. Proof. Setting p = 1 and q = 0, that is,



−a1 2 2 = 1 − λt a F ; −; λt , − t) − t) (1 (z 1 1 0

(2.145)

we deduce the result. Theorem 2.51. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then we have  0

z

t α−1 (z − t)β−1 dt (1 − λt (z − t))a1

= B (α, β) z

α+β−1

3 F2

1 2 , α+β α+β+1 4 λz ; 2 , 2 a1 , α, β;

(2.146)

where λ is a constant. Proof. Recalling that p = 1 and q = 0, that is, −a1 1 F0 (a1 ; −; λt (1 − t)) = (1 − λt (z − t))

we obtain the result.

,

(2.147)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 41   Theorem 2.52. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have 

z

  t α−1 (z − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt

0

= B (α, β) zα+β−1

a1 , · · · , ap , βs , · · · , β+s−1 ; s × p+s Fq+s λzs , α+β+s−1 , · · · , ; c1 , · · · , cq , α+β s s

(2.148)

where λ is a constant. Proof. On changing the variable to λ (z − t)s = λl s , we get the result. Theorem 2.53. If Re (α) > 0, Re (β) > 0, s ∈ N, and |z| < 1, then we have 

z

s

t α−1 (z − t)β−1 eλ(z−t) dt

0

= B (α, β) zα+β−1

β β+1 β+s−1 ; s , s ,··· , s s λz , × s Fs α+β α+β+1 , · · · , α+β+s−1 ; s , s s

(2.149)

where λ is a constant. Proof. Taking p = 0 and q = 0, that is, 0 F0

  s −; −; λ (z − t)s = eλ(z−t) ,

(2.150)

we conclude the result. In a similar manner, we get the following results. Theorem 2.54. If Re (α) > 0, Re (β) > 0, Re (c1 ) > 0, s ∈ N, and |z| < 1, then we have 

z

  t α−1 (z − t)β−1 0 F1 −; c1 ; λ (z − t)s dt

0

= B (α, β) zα+β−1

β β+1 β+s−1 ; s , s ,··· , s s × s F1+s λz , α+β+1 α+β+s−1 , , · · · , ; c1 , α+β s s s where λ is a constant. Proof. Taking p = 0 and q = 1, we get the result.

(2.151)

42

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 2.55. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, s ∈ N, and |z| < 1, then we have 

z

 −a t α−1 (z − t)β−1 1 − λ (z − t)s 1 dt

0

= B (α, β) zα+β−1

β+s−1 a1 , βs , β+1 ; s ,··· , s s λz , × 1+s Fs α+β α+β+1 , · · · , α+β+s−1 ; s , s s

(2.152)

where λ is a constant. Proof. Recalling that p = 1 and q = 0, that is, 1 F0

   −a a1 ; −; λt (1 − t)s = 1 − λt (z − t)s 1 ,

(2.153)

we get the result. Theorem 2.56. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, s ∈ N, and |z| < 1, then we have  z   t α−1 (z − t)β−1 1 F1 a1 ; c1 ; λ (z − t)s dt 0

= B (α, β) zα+β−1

β+s−1 a1 , βs , β+1 ; s ,··· , s s × 1+s F1+s λz , α+β+1 c1 , α+β , · · · , α+β+s−1 ; s , s s

(2.154)

where λ is a constant. Proof. Putting p = 1 and q = 1, we get the result. Theorem 2.57. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, s ∈ N, and |z| < 1, then we have 

z

  t α−1 (z − t)β−1 1 F2 a1 ; c1 , c2 ; λ (z − t)s dt

0

= B (α, β) zα+β−1

β+s−1 a1 , βs , β+1 ; s ,··· , s s × 1+s F2+s λz , α+β+1 c1 , c2 , α+β , · · · , α+β+s−1 ; s , s s where λ is a constant. Proof. Recalling that p = 1 and q = 2, we have the result.

(2.155)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 43

Theorem 2.58. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, s ∈ N, and |z| < 1, then we have  z   t α−1 (z − t)β−1 2 F1 a1 , a2 ; c1 ; λ (z − t)s dt 0

= B (α, β) zα+β−1

β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s s × 2+s F1+s λz , α+β+1 c1 , α+β , · · · , α+β+s−1 ; s , s s

(2.156)

where λ is a constant. Proof. Taking p = 1 and q = 2, we get the result.

  Theorem 2.59. If Re > 0, Re > 0, Re > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , (α) (β) (a ) 1   Re cq > 0, p, q ∈ N0 , and |z| < 1, then we have  z   t α−1 (z − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t) dt 0   (2.157) a1 , · · · , ap , β; α+β−1 λz , = B (α, β) z p+1 Fq+1 c1 , · · · , cq , α + β; where λ is a constant. Proof. Setting s = 1, we get the result. Theorem 2.60. If Re (α) > 0, Re (β) > 0, and |z| < 1, then we have  z t α−1 (z − t)β−1 eλ(z−t) dt 0   β; α+β−1 = B (α, β) z λz , 1 F1 α + β;

(2.158)

where λ is a constant. Proof. Recalling that p = 0 and q = 0, that is, 0 F0 (−; −; λ (z − t)) = e

λ(z−t)

,

(2.159)

we get the result. Theorem 2.61. If Re (α) > 0, Re (β) > 0, Re (c1 ) > 0, and |z| < 1, then we have  z t α−1 (z − t)β−1 0 F1 (−; c1 ; λ (z − t)) dt 0   β; λz , = B (α, β) zα+β−1 1 F2 c1 , α + β; where λ is a constant.

(2.160)

44

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Proof. In view of p = 0 and q = 1, we deduce the result. Theorem 2.62. If Re (α) > 0, Re (β) > 0, and |z| < 1, then we have  z t α−1 (z − t)β−1 (1 − λz + λt)−a1 dt 0   a1 , β; α+β−1 λz , = B (α, β) z 2 F1 α + β;

(2.161)

where λ is a constant. Proof. Putting p = 1 and q = 0, that is, −a1 1 F0 (a1 ; −; λ (1 − t)) = (1 − λ (z − t))

,

(2.162)

we deduce the result. Theorem 2.63. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, and |z| < 1, then we have  z t α−1 (z − t)β−1 1 F1 (a1 ; c1 ; λ (z − t))dt 0   (2.163) a1 , β; λz , = B (α, β) zα+β−1 2 F2 c1 , α + β; where λ is a constant. Proof. Setting p = 1 and q = 1, we have the result. Theorem 2.64. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, and |z| < 1, then we have  z t α−1 (z − t)β−1 1 F2 (a1 ; c1 , c2 ; λ (z − t))dt 0   (2.164) a1 , β; λz , = B (α, β) zα+β−1 2 F3 c1 , c2 , α + β; where λ is a constant. Proof. Making use of p = 1 and q = 2, we have the result. Theorem 2.65. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and |z| < 1, then we have  z t α−1 (z − t)β−1 2 F1 (a1 , a2 ; c1 ; λ (z − t))dt 0   (2.165) a1 , a2 , β; λz , = B (α, β) zα+β−1 3 F2 c1 , α + β; where λ is a constant.

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 45

Proof. On account of p = 2 and q = 1, we deduce to the result. Theorem 2.66. If Re (α) > 0, Re (β) > 0, and |z| < 1, then we have  z 2 t α−1 (z − t)β−1 eλ(z−t) dt 0 β β+1

2, 2 ; α+β−1 2 = B (α, β) z , 2 F2 α+β α+β+1 λz , ; 2 2

(2.166)

where λ is a constant. Proof. Putting p = 0 and q = 0, that is,

2 2 = eλ(z−t) , 0 F0 −; −; λ (z − t)

(2.167)

we deduce the result. Theorem 2.67. If Re (α) > 0, Re (β) > 0, and |z| < 1, then we have 



z

t

α−1

(z − t)

β−1 z−t

e

dt = B (α, β) z

α+β−1

1 F1

0

 β; z . α + β;

(2.168)

Proof. Setting p = 0 and q = 0, that is, 0 F0 (−; −; z − t) = e

z−t

(2.169)

,

we have the result. Theorem 2.68. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then we have 

z

t

α−1

(z − t)

β−1

−a1

(1 − z + t)

 dt = B (α, β) z

α+β−1

2 F1

0

 a1 , β; z . α + β;

(2.170)

Proof. Recalling that p = 1 and q = 0, that is, 1 F0 (a1 ; −; 1 − t) = (1 − z + t)

−a1

,

(2.171)

we deduce the result. Theorem 2.69. If Re (α) > 0, Re (β) > 0, and |z| < 1, then we have  z 2 t α−1 (z − t)β−1 e(z−t) dt 0 β β+1

2, 2 ; α+β−1 2 = B (α, β) z . 2 F2 α+β α+β+1 z ; 2 , 2

(2.172)

46

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Proof. With the aid of p = 0 and q = 0, that is,

2 2 = e(z−t) , 0 F0 −; −; (z − t)

(2.173)

we get the result.     Let Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, n, p, q ∈ N0 , and |z| < 1. Then we have ((a) , (c) ; λz)   = p Fq a1 , · · · , ap ; c1 , · · · , cq ; λz   a 1 , · · · , ap ; λz = p Fq c1 , · · · , cq   ∞  (a1 )n · · · ap n (λz)n   = , (c1 )n · · · cq n n!

(2.174)

((a) , (c) ; iλz)   = p Fq a1 , · · · , ap ; c1 , · · · , cq ; iλz   a1 , · · · , ap = p Fq ; iλz c1 , · · · , cq   ∞  (a1 )n · · · ap n (iλz)n   , = (c1 )n · · · cq n n!

(2.175)

((a) , (c) ; −iλz)   = p Fq a1 , · · · , ap ; c1 , · · · , cq ; −iλz   a , · · · , ap = p Fq 1 ; −iλz c1 , · · · , cq   ∞  (a1 )n · · · ap (−iλz)n  n , = n! (c1 )n · · · cq n

(2.176)

((a) , (c) ; −λz)   = p Fq a1 , · · · , ap ; c1 , · · · , cq ; −λz   a , · · · , ap = p Fq 1 ; −λz c1 , · · · , cq   ∞  (a1 )n · · · ap (−λz)n  n , = (c1 )n · · · cq n n!

(2.177)

p Fq

n=0

which leads to p Fq

n=0

p Fq

n=0

and p Fq

n=0

√ where i = −1 and λ ∈ C.

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 47

2.1.2 Laplace and Mellin transforms for the Clausen hypergeometric series In this section, we introduce the Laplace and Mellin transforms for the Clausen hypergeometric series and study the identities involving the Clausen hypergeometric series. Theorem 2.70 (Laplace transforms for the Clausen hypergeometric series). Let an , cn , z, λ ∈ C and n, p, q ∈ N0 . Then the Laplace transforms of the function t λ−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; zt is as follows:    L t λ−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; zt    a 1 , · · · , ap λ−1 =L t ; zt p Fq c1 , · · · , cq     ∞ (2.178) a , · · · , ap e−st t λ−1 p Fq 1 ; zt dt = c1 , · · · , cq 0    (λ) λ, a1 , · · · , ap z , = λ p+1 Fq ; c1 , · · · , cq s s where the Laplace transform of a function f is [84]  ∞ e−st f (t) dt. L [f (t)] = f (s) =

(2.179)

0

The convergence cases are as follows: 1. Re (λ) > 0, p ≤ q; 2. Re (s) > 0 if p < q; 3. Re (s) > Re (z) if p = q. The result was obtained by Erdelyi et al. in 1954 [85] at z = 1 and developed by Srivastava and Manocha [22] at the present case. Theorem 2.71 (Mellin transforms for the Clausen hypergeometric series). Let an , cn , z ∈ C, p, q ∈ N0 , z = 0, and 0 < Re ( ) 0; 2. p = q, |arg (z)| ≤ π2 ; 3. p = q + 1, |arg (z)| < π2 . The result was obtained by Srivastava and Kashyap in 1982 [51], and its basic formula was derived from Barnes in 1935 [60]. Theorem 2.72. Let an , cn , z ∈ C and p, q ∈ N0 . Then we have      ∞  (λ) a , · · · , ap λ, a1 , · · · , ap z t λ−1 e−γ t p Fq 1 ; zt dt = λ p+1 Fq ; , c1 , · · · , cq c1 , · · · , cq γ γ 0

(2.182)

where γ ∈ C, and the convergence cases are as follows: 1. Re (λ) > 0, p ≤ q; 2. Re (γ ) > 0 if p < q; 3. Re (γ ) > Re (z) if p = q. Example 2.1. 

∞

t λ−1 e−γ t ezt dt =

0

Example 2.2.



∞

t

λ−1 −γ t

e

0

Example 2.3.

  z λ  (λ) 1 − . γλ γ

   (λ) λ, a1 z . ; (1 − zt) dt = λ 2 F0 − γ γ



a1

∞

t λ−1 e−2t dt = 2λ  (λ) .

(2.183)

(2.184)

(2.185)

0

Theorem 2.73. Let an , cn , z ∈ C and p, q ∈ N0 . Then we have      ∞ a , · · · , ap λ, a1 , · · · , ap t λ−1 e−t p Fq 1 ; zt dt =  (λ) p+1 Fq ;z , c1 , · · · , cq c1 , · · · , cq 0

(2.186)

and the convergence cases are as follows: 1. p ≤ q; 2. 1 > Re (z) if p = q. Theorem 2.74. Let an , cn ∈ C and p, q ∈ N0 . Then we have      ∞  (λ) a , · · · , ap λ, a1 , · · · , ap 1 , t λ−1 e−γ t p Fq 1 ; t dt = λ p+1 Fq ; c1 , · · · , cq c1 , · · · , cq γ γ 0 where γ ∈ C, and the convergence cases are as follows:

(2.187)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 49

1. Re (λ) > 0, p ≤ q; 2. Re (γ ) > 0 if p < q; 3. Re (γ ) > Re (z) if p = q. Theorem 2.75. Let an , cn ∈ C and p, q ∈ N0 . Then we have 

∞

t λ−1 e−t p Fq



0

   a 1 , · · · , ap λ, a1 , · · · , ap ; t dt =  (λ) p+1 Fq ;1 , c1 , · · · , cq c1 , · · · , cq

(2.188)

where the convergence cases are as follows: 1. γ ∈ C and Re (λ) > 0 if p ≤ q; 2. γ ∈ C if p < q. Theorem 2.76. Let an , cn , z, λ ∈ C, p, q ∈ N0 , z = 0, and 0 < Re (λ) < min1≤i≤p Re (ai ). Then we have 

∞

t

λ−1

0

  q p   a1 , · · · , ap i=1  (ai − ) i=j  cj   · z−λ , ; −zt dt =  (ω) · p q p Fq c1 , · · · , cq   c −

(a ) i j i=j i=1

(2.189)

where the convergence cases are as follows: 1. p = q − 1, z > 0; 2. p = q, |arg (z)| ≤ π2 ; 3. p = q + 1, |arg (z)| < π2 . Theorem 2.77. Let an , cn , λ ∈ C, p, q ∈ N0 , and 0 < Re (λ) < min1≤i≤p Re (ai ). Then we have 



∞

t 0

λ−1

p Fq

  q p   (λ) i=1  (ai − λ) i=j  cj a 1 , · · · , ap   , ; −t dt = p q c1 , · · · , cq i=1  (ai ) i=j  cj − λ

(2.190)

where the convergence cases are as follows: 1. p = q − 1; 2. p = q.

2.2 The hypergeometric supertrigonometric functions via Clausen hypergeometric series In this section, we propose the hypergeometric supertrigonometric functions via Clausen hypergeometric series and give the convergence, properties, and theorems for the new special functions.

50

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

2.2.1 Definitions, convergence, properties, and theorems for the hypergeometric supertrigonometric functions via Clausen hypergeometric series On account of the decomposition scheme for the complex exponential function proposed by Euler in 1748 [57], we present this scheme to establish the theory of the hypergeometric supertrigonometric functions via Clausen hypergeometric series. Definition 2.2. The hypergeometric supersine via Clausen hypergeometric series is defined as   p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; z   a , · · · , ap = p Supersinq 1 ;z c1 , · · · , cq (2.191)   ∞ n 2n+1  (a1 )2n+1 · · · ap 2n+1 (−1) z   , = (c1 )2n+1 · · · cq 2n+1 (2n + 1)! n=0

where an , cn ∈ C and n, p, q ∈ N0 . From Eq. (2.91) we find that

  (a1 )1 · · · ap 1   z, n = 0 : 0 (z) = (c1 )1 · · · cq 1   (a1 )3 · · · ap 3 z3   , n = 1 : 1 (z) = − (c1 )3 · · · cq 3 3!   (a1 )5 · · · ap 5 z5   n = 2 : 2 (z) = , (c1 )5 · · · cq 5 5!   (a1 )7 · · · ap 7 z7   n = 3 : 3 (z) = − , (c1 )7 · · · cq 7 7!   (a1 )9 · · · ap 9 z9   , n = 4 : 4 (z) = (c1 )9 · · · cq 9 9!   (a1 )11 · · · ap 11 z11   n = 5 : 5 (z) = − , (c1 )11 · · · cq 11 11!

(2.192)

(2.193)

(2.194)

(2.195)

(2.196)

(2.197)

and so on. The convergence for the hypergeometric supersine via Clausen hypergeometric series is as follows: 1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, q p it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0.

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 51

Definition 2.3. The hypergeometric supercosine via Clausen hypergeometric series is defined as   p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; z   a , · · · , ap ;z = p Supercos q 1 c1 , · · · , cq (2.198)   ∞ n 2n  (a1 )2n · · · ap (−1) z   2n = , (c1 )2n · · · cq 2n (2n)! n=0

where an , cn ∈ C and n, p, q ∈ N0 . From Eq. (2.198) we have that n = 0 : 0 (z) = 1,   (a1 )2 · · · ap 2 z2   n = 1 : 1 (z) = − , (c1 )2 · · · cq 2 2!   (a1 )4 · · · ap 4 z4   , n = 2 : 2 (z) = (c1 )4 · · · cq 4 4!   (a1 )6 · · · ap 6 z6   , n = 3 : 3 (z) = − (c1 )6 · · · cq 6 6!   (a1 )8 · · · ap 8 z8   n = 4 : 4 (z) = , (c1 )8 · · · cq 8 8!   (a1 )10 · · · ap 10 z10   n = 5 : 5 (z) = − , (c1 )10 · · · cq 10 10!

(2.199) (2.200)

(2.201)

(2.202)

(2.203)

(2.204)

and so on. The convergence for the hypergeometric supercosine via Clausen hypergeometric series is given as follows: 1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, q p it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0. Definition 2.4. The hypergeometric supertangent via Clausen hypergeometric series is defined as     p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; z  , (2.205) p Supertanq a1 , · · · , ap ; c1 , · · · , cq ; z = p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; z where an , cn ∈ C and p, q ∈ N0 .

52

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

The cases of convergence for the hypergeometric supercosine via Clausen hypergeometric series are as follows: 1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, p q it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0. Definition 2.5. The hypergeometric supercotangent via Clausen hypergeometric series is defined as     p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; z  , (2.206) p Supercot q a1 , · · · , ap ; c1 , · · · , cq ; z = p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; z where an , cn ∈ C and p, q ∈ N0 . The cases of convergence for the hypergeometric supercosine via Clausen hypergeometric series are given as follows: 1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, p q it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0. Definition 2.6. The hypergeometric supersecant via Clausen hypergeometric series is defined as p Supersec q

  a1 , · · · , ap ; c1 , · · · , cq ; z =

1  , p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; z

(2.207)

where an , cn ∈ C and p, q ∈ N0 . The cases of convergence for the hypergeometric supercosine via Clausen hypergeometric series are given as follows: 1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, p q it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0. Definition 2.7. The hypergeometric supercosecant via Clausen hypergeometric series is defined as p Supercosec q

  a1 , · · · , ap ; c1 , · · · , cq ; z =

1  , a Supersin , · · · , ap ; c1 , · · · , cq ; z p q 1

(2.208)

where an , cn ∈ C and p, q ∈ N0 . The cases of convergence for the hypergeometric supercosine via Clausen hypergeometric series are given as follows:

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 53

1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, p q it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0. Definition 2.8. The family of the containing the hyper hypergeometric functions  geometric supersine p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; z , hypergeometric supercosine  p Supercos q  a1 , · · · , ap ; c1 , · · · , cq ; z , hypergeometric supertangent p Supertanq a1, · · · , ap ;  c1 , · · · , cq ; z , hypergeometric supercotangent p Supercot q a1 , · · · , ap ; c1 , · · · , cq ; z , hyper geometric supersecant p Supersec q a1 , · · · , ap; c1 , · · · , cq ; z , and hypergeometric supercosecant p Supercosec q a1 , · · · , ap ; c1 , · · · , cq ; z is called the hypergeometric supertrigonometric functions via Clausen hypergeometric series. Now we consider the hypergeometric supersine, hypergeometric supercosine, hypergeometric supertangent, hypergeometric supercotangent, hypergeometric supersecant, and hypergeometric supercosecant with λ ∈ C. We easily see that   p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λz     1  = p Fq a1 , · · · , ap ; c1 , · · · , cq ; iλz − p Fq a1 , · · · , ap ; c1 , · · · , cq ; −iλz 2i (2.209)   ∞  (a1 )2n+1 · · · ap 2n+1 (−1)n (λz)2n+1   , = (2n + 1)! (c1 )2n+1 · · · cq 2n+1 n=0   p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λz     1 = p Fq a1 , · · · , ap ; c1 , · · · , cq ; iλz + p Fq a1 , · · · , ap ; c1 , · · · , cq ; −iλz 2 (2.210)   ∞  (a1 )2n · · · ap 2n (−1)n (λz)2n   , = (2n)! (c1 )2n · · · cq 2n n=0     p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λz   , (2.211) p Supertanq a1 , · · · , ap ; c1 , · · · , cq ; λz = p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λz     p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λz   , (2.212) p Supercot q a1 , · · · , ap ; c1 , · · · , cq ; λz = p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λz p Supersec q

  a1 , · · · , ap ; c1 , · · · , cq ; λz =

1  , a Supercos , · · · , ap ; c1 , · · · , cq ; λz p q 1

(2.213)

and p Supercosec q

  a1 , · · · , ap ; c1 , · · · , cq ; λz =

where an , cn , λ ∈ C and n, p, q ∈ N0 , i =

√

1   , (2.214) p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λz

−1.

54

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

On account of the above results, we take into consideration that   p Fq a1 , · · · , ap ; c1 , · · · , cq ; iz     = p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; z + i p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; z ,

(2.215)

where an , cn ∈ C and p, q ∈ N0 .

2.2.2 The series of the hypergeometric supertrigonometric functions via Clausen hypergeometric series In some considerations, we observe the families of the hypergeometric series that are based on the hypergeometric supertrigonometric functions via Clausen hypergeometric series. Starting from the idea, we may find a class of the series for the hypergeometric supertrigonometric functions via Clausen hypergeometric series as follows. Series 1     For Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, κ, p, q ∈ N0 , and |z| < 1, we consider the series of the form  (z) =

∞ 

φ (κ)



p Fq

  a1 , · · · , ap ; c1 , · · · , cq ; iκz ,

(2.216)

κ=0

which can be represented in the form  (z) ∞      =γ + ϕ (κ) p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; κz κ=1

+

∞ 

(2.217)

    ψ (κ) p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; κz ,

κ=1

where ϕ (0) − iψ (0) , 2 ϕ (κ) − iψ (κ) φ (κ) = , 2 γ=

(2.218) (2.219)

and ϕ (κ) + iψ (κ) (2.220) 2 are the coefficients of the supertrigonometric functions via Clausen hypergeometric series. In the present case, we can suggest the particular cases of the supertrigonometric functions via Clausen hypergeometric series as follows: φ (−κ) =

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 55

Series 2 If  (−z) =  (z), then there is the series of the form  (z) = γ +

∞      ϕ (κ) p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; κz ,

(2.221)

κ=1

where γ and ϕ (κ) are the coefficients of the supertrigonometric functions via Clausen hypergeometric series. Series 3 If  (−z) = − (z), then the series can be taken in the form  (z) =

∞      ψ (κ) p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; κz ,

(2.222)

κ=1

where ψ (κ) are the coefficients of the supertrigonometric functions via Clausen hypergeometric series. On account of the definitions of the hypergeometric supertrigonometric functions, we reduce to the properties as follows. Property 2.4. Suppose that an , cn ∈ C and p, q ∈ N0 . Then     a1 , · · · , ap ; c1 , · · · , cq ; −z = −p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; z ,     p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; −z = p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; z ,     p Supertanq a1 , · · · , ap ; c1 , · · · , cq ; −z = −p Supertanq a1 , · · · , ap ; c1 , · · · , cq ; z ,     p Supercot q a1 , · · · , ap ; c1 , · · · , cq ; −z = −p Supercot q a1 , · · · , ap ; c1 , · · · , cq ; z ,     p Supersec q a1 , · · · , ap ; c1 , · · · , cq ; −z = p Supersec q a1 , · · · , ap ; c1 , · · · , cq ; z , p Supersinq

(2.223) (2.224) (2.225) (2.226) (2.227)

and p Supercosec q

    a1 , · · · , ap ; c1 , · · · , cq ; −z = −p Supercosec q a1 , · · · , ap ; c1 , · · · , cq ; z . (2.228)

In the same manner as for the Clausen hypergeometric series, we conclude the following theorems.     Theorem 2.78. Suppose that Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and p, q ∈ N. Then we have   p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λz  1 (2.229)   t a1 −1 (1 − t)c1 −a1 −1 p−1 Supersinq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt, = 0

56

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

where

  a2 , · · · , ap ; c2 , · · · , cq ; λzt     1  = p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; iλz − p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; −iλz . 2i (2.230)

p−1 Supersinq−1

Proof. By the equalities   p Fq a1 , · · · , ap ; c1 , · · · , cq ; iλz  1   t a1 −1 (1 − t)c1 −a1 −1 p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; iλzt dt =

(2.231)

0

and p Fq



  a1 , · · · , ap ; c1 , · · · , cq ; −iλz

=

1

  t a1 −1 (1 − t)c1 −a1 −1 p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; −iλzt dt

(2.232)

0

we obtain the equality   p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λz     1  = p Fq a1 , · · · , ap ; c1 , · · · , cq ; iλz − p Fq a1 , · · · , ap ; c1 , · · · , cq ; −iλz 2i  1   = t a1 −1 (1 − t)c1 −a1 −1 p−1 Supersinq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt

(2.233)

0

since

  a2 , · · · , ap ; c1 , · · · , cq ; λzt     (2.234) 1  = p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; iλz − p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; −iλz 2i

p−1 Supersinq−1

where Re (c1 ) > Re (a1 ) > 0, and λ ∈ C.

    Theorem 2.79. Suppose that Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and p, q ∈ N. Then we have   p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λz  1 (2.235)   t a1 −1 (1 − t)c1 −a1 −1 p−1 Supercos q−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt, = 0

where

  a2 , · · · , ap ; c1 , · · · , cq ; λzt     (2.236) 1 = p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; iλz + p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; −iλz . 2

p−1 Supercos q−1

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 57

Proof. By the relations   p Fq a1 , · · · , ap ; c1 , · · · , cq ; iλz  1   t a1 −1 (1 − t)c1 −a1 −1 p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; iλzt dt =

(2.237)

0

and p Fq



  a1 , · · · , ap ; c1 , · · · , cq ; −iλz

=

1

  t a1 −1 (1 − t)c1 −a1 −1 p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; −iλzt dt

(2.238)

0

we obtain

  a1 , · · · , ap ; c1 , · · · , cq ; λz     1 = p Fq a1 , · · · , ap ; c1 , · · · , cq ; iλz + p Fq a1 , · · · , ap ; c1 , · · · , cq ; −iλz 2  1   t a1 −1 (1 − t)c1 −a1 −1 p−1 Supercos q−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt, =

p Supercos q

(2.239)

0

where

  a2 , · · · , ap ; c1 , · · · , cq ; λzt     (2.240) 1 = p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; iλz + p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; −iλz 2     with Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and p, q ∈ N0 . p−1 Supercos q−1

2.2.3 Laplace transforms for the hypergeometric supertrigonometric functions via Clausen hypergeometric series and related results In this section, we propose the Laplace transforms for the hypergeometric supertrigonometric functions via Clausen hypergeometric series and present the particular cases. Theorem 2.80. Let an , cn , z, λ ∈ C and n, p, q ∈ N0 .  λ−1 Supersin a , · · · , a ; c , · · · , c ; q 1 p 1 q p  Then there the Laplace transforms of the function t zt is as follows:    L t λ−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; zt (2.241) z  (λ) , = λ p+1 Supersinq λ, a1 , · · · , ap ; c1 , · · · , cq ; s s where the convergence cases are as follows: 1. Re (λ) > 0, p ≤ q; 2. Re (s) > 0 if p < q; 3. Re (s) > Re (z) if p = q.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Proof. Using the equalities    L t λ−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; izt     ∞ a 1 , · · · , ap −st λ−1 t = e ; izt dt p Fq c1 , · · · , cq 0    (λ) λ, a1 , · · · , ap iz = λ p+1 Fq ; c1 , · · · , cq s s

(2.242)

   L t λ−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; −izt     ∞ a , · · · , ap = e−st t λ−1 p Fq 1 ; −izt dt c1 , · · · , cq 0    (λ) λ, a1 , · · · , ap iz , = λ p+1 Fq ;− c1 , · · · , cq s s

(2.243)

and

we have the result. Theorem 2.81. Let an , cn , z, λ ∈ C and n, p, q ∈ N0 .   Then the Laplace transform of the function t λ−1 p Supercosinq a1 , · · · , ap ; c1 , · · · , cq ; zt is as follows:    L t λ−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; zt  (λ) z , = λ p+1 Supercos q λ, a1 , · · · , ap ; c1 , · · · , cq ; s s

(2.244)

where the convergent cases are as follows: 1. Re (λ) > 0, p ≤ q; 2. Re (s) > 0 if p < q; 3. Re (s) > Re (z) if p = q. Proof. Using the identities    L t λ−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; izt     ∞ a , · · · , ap = e−st t λ−1 p Fq 1 ; izt dt c1 , · · · , cq 0    (λ) λ, a1 , · · · , ap iz = λ p+1 Fq ; c1 , · · · , cq s s

(2.245)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 59

and

   L t λ−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; −izt     ∞ a , · · · , ap = e−st t λ−1 p Fq 1 ; −izt dt c1 , · · · , cq 0    (λ) λ, a1 , · · · , ap iz , = λ p+1 Fq ;− c1 , · · · , cq s s

(2.246)

we deduce the result.     Theorem If p ≤ q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, q 2.82. p Re k=1 ck − k=1 ak > 0, and |z| < 1, then   a1 , · · · , ap ; c1 , · · · , cq ; λz  (c1 ) =  (a1 )  (c1 − a1 )  1   × t a1 −1 (1 − t)c1 −a1 −1 p−1 Supersinq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt,

p Supersinq

(2.247)

0

where λ is a constant.     Theorem If p ≤ q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, q 2.83. p Re k=1 ck − k=1 ak > 0, and |z| < 1, then we have  p+1 Supersinq+1

a1 , · · · , ap , a; λz c1 , · · · , cq , c;



 (c)  (a)  (c − a)  1   × t a−1 (1 − t)c−a−1 p Supersinq a1 , a2 , · · · , ap ; c1 , c2 , · · · , cq ; λzt dt, =

(2.248)

0

where λ is a constant.   Theorem 2.84. If Re(α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,    q p Re cq > 0, |z| < 1, Re k=1 ck − k=1 ak > 0, and κ ∈ N, then 1 B (α, β)



z

  t α−1 (z − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt

0

= zα+β−1 × p+κ Supersinq+κ where λ is a constant.



α+κ−1 a1 , · · · , ap , ακ , α+1 ; κ ,··· , κ

c1 , · · ·

α+β+1 , cq , α+β ,··· κ , κ

,

α+β+κ−1 ; κ

λzκ ,

(2.249)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

  Theorem 2.85. If Re (α) > 0, Re(β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   q p Re cq > 0, Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

  t α−1 (z − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; α+β−1 λz , × p+1 Supersinq+1 =z c1 , · · · , cq , α + β;

1 B (α, β)

z

(2.250)

where λ is a constant.   Theorem 2.86. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   p q Re cq > 0, and Re k=1 ck − k=1 ak > 0, then 

  t α−1 (1 − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; λ , = p+1 Supersinq+1 c1 , · · · , cq , α + β;

1 B (α, β)

1

(2.251)

where λ is a constant.   Theorem 2.87. If Re > 0, Re > 0, Re > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , (α) (β) (a ) 1   Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have  z   t α−1 (z − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt 0

= B (α, β) zα+β−1



× p+κ+s Supersinq+κ+s

, βs , · · · , β+s−1 ; a1 , · · · , ap , ακ , · · · , α+κ−1 κ s α+β α+β+κ+s−1 c1 , · · · , cq , κ+s , · · · , ; κ+s

κ κ s s λzκ+s (κ + s)κ+s



(2.252)

,

where λ is a constant.   Theorem 2.88. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   t α−1 (1 − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt 0

= B (α, β) × p+κ+s Supersinq+κ+s



, βs , · · · , β+s−1 ; a1 , · · · , ap , ακ , · · · , α+κ−1 κ s α+β α+β+κ+s−1 ; c1 , · · · , cq , κ+s , · · · , κ+s

κ κ ss λ (κ + s)κ+s



(2.253)

,

where λ is a constant.   Theorem 2.89. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 61

1 B (α, β)



  (1 + t)α−1 (1 − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt

1

−1

= 2α+β−1



× p+κ+s Supersinq+κ+s

, βs , · · · , β+s−1 ; κ κ s s λ2κ+s a1 , · · · , ap , ακ , · · · , α+κ−1 κ s α+β+κ+s−1 (κ + s)κ+s ; c1 , · · · , cq , α+β κ+s , · · · , κ+s

, (2.254)

where λ is a constant.   Theorem 2.90. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have  z   1 t α−1 (z − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt B (α, β) 0 = zα+β−1



× p+s Supersinq+s

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s α+β+1 , · · · , α+β+s−1 ; c1 , · · · , cq , α+β s , s s



(2.255)

λzs ,

where λ is a constant.   Theorem 2.91. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  t α−1 (1 − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s = p+s Supersinq+s λ , α+β+1 , · · · , α+β+s−1 ; c1 , · · · , cq , α+β s , s s

1 B (α, β)

1

(2.256)

where λ is a constant.   Theorem 2.92. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 1 B (α, β)



1 −1

  (1 + t)α−1 (1 − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt

= 2α+β−1 × p+s Supersinq+s where λ is a constant.



(2.257)



β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s 2s λ α+β α+β+1 α+β+s−1 ; c1 , · · · , cq , s , s , · · · , s

,

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

    Theorem 2.93. If p ≤ q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, q  p Re k=1 ck − k=1 ak > 0, and |z| < 1, then   a1 , · · · , ap ; c1 , · · · , cq ; λz  (c1 ) =  (a1 )  (c1 − a1 )  1   × t a1 −1 (1 − t)c1 −a1 −1 p−1 Supercos q−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt,

p Supercos q

(2.258)

0

where λ is a constant.     Theorem 2.94. If p ≤ q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0,  q p Re k=1 ck − k=1 ak > 0, and |z| < 1, then we have  p+1 Supercos q+1

a1 , · · · , ap , a; λz c1 , · · · , cq , c;



 (c)  (a)  (c − a)  1   × t a−1 (1 − t)c−a−1 p Supercos q a1 , a2 , · · · , ap ; c1 , c2 , · · · , cq ; λzt dt, =

(2.259)

0

where λ is a constant.   Theorem 2.95. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , q    p Re cq > 0, |z| < 1, Re k=1 ck − k=1 ak > 0, and κ ∈ N, then 1 B (α, β)



z

  t α−1 (z − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt

0

= zα+β−1



× p+κ Supercos q+κ

α+κ−1 ; a1 , · · · , ap , ακ , α+1 κ ,··· , κ κ α+β α+β+1 α+β+κ−1 λz ; c1 , · · · , cq , κ , κ , · · · , κ



(2.260)

,

where λ is a constant.   Theorem 2.96. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , q    p Re cq > 0, Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

  t α−1 (z − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; α+β−1 λz , × p+1 Supercos q+1 =z c1 , · · · , cq , α + β;

1 B (α, β)

z

where λ is a constant.

(2.261)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 63   Theorem 2.97. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   q p Re cq > 0, and Re k=1 ck − k=1 ak > 0, then 

  t α−1 (1 − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; λ , = p+1 Supercos q+1 c1 , · · · , cq , α + β;

1 B (α, β)

1

(2.262)

where λ is a constant.

  Theorem 2.98. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have  z   t α−1 (z − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt 0

= B (α, β) zα+β−1



× p+κ+s Supercos q+κ+s

, βs , · · · , β+s−1 ; a1 , · · · , ap , ακ , · · · , α+κ−1 κ s α+β α+β+κ+s−1 ; c1 , · · · , cq , κ+s , · · · , κ+s

κ κ s s λzκ+s



(2.263)

,

(κ + s)κ+s

where λ is a constant.

  Theorem 2.99. If Re > 0, Re > 0, Re > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , (α) (β) (a ) 1   Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have 

1

  t α−1 (1 − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt

0

= B (α, β)



× p+κ+s Supercos q+κ+s

, βs , · · · , β+s−1 ; a1 , · · · , ap , ακ , · · · , α+κ−1 κ s α+β α+β+κ+s−1 c1 , · · · , cq , κ+s , · · · , ; κ+s



κ κ ss λ

(2.264)

,

(κ + s)κ+s

where λ is a constant.

  Theorem 2.100. If Re > 0, Re > 0, Re > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , (α) (β) (a ) 1   Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have 1 B (α, β) =2



1

−1

  (1 + t)α−1 (1 − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt

α+β−1

× p+κ+s Supercos q+κ+s



, βs , · · · , β+s−1 ; κ κ s s λ2κ+s a1 , · · · , ap , ακ , · · · , α+κ−1 κ s α+β+κ+s−1 (κ + s)κ+s c1 , · · · , cq , α+β ; κ+s , · · · , κ+s

, (2.265)

where λ is a constant.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

  Theorem 2.101. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have 1 B (α, β)



z

  t α−1 (z − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt

0

= zα+β−1



× p+s Supercos q+s

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s

c1 , · · ·

α+β+1 , cq , α+β ,··· s , s

,

α+β+s−1 ; s

(2.266)

λzs ,

where λ is a constant.   Theorem 2.102. If Re > 0, Re > 0, Re > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , (α) (β) (a ) 1   Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  t α−1 (1 − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s = p+s Supercos q+s λ , α+β+1 c1 , · · · , cq , α+β , · · · , α+β+s−1 ; s , s s

1 B (α, β)

1

(2.267)

where λ is a constant.   Theorem 2.103. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 1 B (α, β) =2



1 −1

  (1 + t)α−1 (1 − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt

α+β−1



× p+s Supercos q+s

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s 2s λ α+β α+β+1 α+β+s−1 ; c1 , · · · , cq , s , s , · · · , s

(2.268)

,

where λ is a constant.     Corollary If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, |z| < 1,  q 2.7.  p Re k=1 ck − k=1 ak > 0, and κ ∈ N, then 1 B (α, 1) = zα



z

  t α−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt

0

× p+κ Supersinq+κ where λ is a constant.



a1 , · · · , ap , κ1 , κ2 , · · · , 1; λzκ α+1 α+2 α+κ c1 , · · · , cq , κ , κ , · · · , κ ;

(2.269)

,

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 65   Corollary 2.8. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   q  p Re cq > 0, |z| < 1, Re k=1 ck − k=1 ak > 0, and κ ∈ N, then 1 B (1, β)



z

  (z − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt

0

= zβ



× p+κ Supersinq+κ

a1 , · · · , ap , κ1 , κ2 , · · · , 1; c1 , · · ·

2+β , cq , 1+β κ , κ ,···

,

β+κ λz κ ;

κ

(2.270)

,

where λ is a constant.     Corollary 2.9. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, q  p Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

  t α−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; α λz , = z × p+1 Supersinq+1 c1 , · · · , cq , α + 1;

1 B (α, 1)

z

(2.271)

where λ is a constant.     Corollary 2.10. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, q  p Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

  (z − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , 1; β λz , = z × p+1 Supersinq+1 c1 , · · · , cq , 1 + β;

1 B (1, β)

z

(2.272)

where λ is a constant.     Corollary 2.11. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and q  p Re k=1 ck − k=1 ak > 0, then 

  t α−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; λ , = p+1 Supersinq+1 c1 , · · · , cq , α + 1;

1 B (α, 1)

where λ is a constant.

1

(2.273)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

    Corollary 2.12. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and q  p Re k=1 ck − k=1 ak > 0, then 

  (1 − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , 1; λ , = p+1 Supersinq+1 c1 , · · · , cq , 1 + β;

1 B (1, β)

1

(2.274)

where λ is a constant.   Corollary 2.13. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have 

z

  t α−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt

0

= B (α, 1) zα



× p+κ+s Supersinq+κ+s

, 1s , · · · a1 , · · · , ap , ακ , · · · , α+κ−1 κ α+κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

, 1;

κ κ s s λzκ+s

(2.275)

,

(κ + s)κ+s

where λ is a constant.     Corollary 2.14. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have  z   (z − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt 0

= B (1, β) zβ



× p+κ+s Supersinq+κ+s

; κ κ s s λzκ+s a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 s , β+κ+s (κ + s)κ+s c1 , · · · , cq , 1+β κ+s , · · · , κ+s ;

(2.276)

where λ is a constant.     Corollary 2.15. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have 

1

  t α−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt

0

= B (α, 1) × p+κ+s Supersinq+κ+s where λ is a constant.



, 1s , · · · , 1; a1 , · · · , ap , ακ , · · · , α+κ−1 κ

κ κ ss λ

α+κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

(κ + s)κ+s

(2.277)

,

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 67     Corollary 2.16. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   (1 − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt 0

= B (1, β)



× p+κ+s Supersinq+κ+s

; a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 s 1+β β+κ+s c1 , · · · , cq , κ+s , · · · , κ+s ;



(2.278)

κ κ ss λ , (κ + s)κ+s

where λ is a constant.

    Corollary 2.17. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   1 (1 + t)α−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt B (α, 1) −1 = 2α



× p+κ+s Supersinq+κ+s

, 1s , · · · , ss ; a1 , · · · , ap , ακ , · · · , α+κ−1 κ α+κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

κ κ s s λ2κ+s (κ + s)κ+s

(2.279)

,

where λ is a constant.

    Corollary 2.18. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   1 (1 − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt B (1, β) −1 = 2β



; κ κ s s λ2κ+s a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 s × p+κ+s Supersinq+κ+s , β+κ+s (κ + s)κ+s c1 , · · · , cq , 1+β κ+s , · · · , κ+s ;

(2.280)

where λ is a constant.

    Corollary 2.19. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have  z   1 t α−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt B (α, 1) 0 = zα × p+s Supersinq+s where λ is a constant.



a1 , · · · , ap , 1s , 2s , · · · , 1; λzs α+2 α+s c1 , · · · , cq , α+1 , , · · · , ; s s s

,

(2.281)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

    Corollary 2.20. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have 1 B (1, β)



z

  (z − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt

0

= zβ × p+s Supersinq+s

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s c1 , · · ·

2+β , cq , 1+β s , s ,···

,

β+s s ;



(2.282)

λzs ,

where λ is a constant.     Corollary 2.21. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  t α−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

a1 , · · · , ap , 1s , 2s , · · · , 1; λ , = p+s Supersinq+s α+2 α+s c1 , · · · , cq , α+1 s , s ,··· , s ;

1 B (α, 1)

1

(2.283)

where λ is a constant.     Corollary 2.22. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  (1 − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s = p+s Supersinq+s λ , β+2 β+s c1 , · · · , cq , 1+β s , s ,··· , s ;

1 B (1, β)

1

(2.284)

where λ is a constant.     Corollary 2.23. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 1 B (α, 1) =2



1

−1

  (1 + t)α−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt

α

× p+s Supersinq+s where λ is a constant.



a1 , · · · , ap , 1s , 2s , · · · , 1; 2s λ α+2 α+s c1 , · · · , cq , α+1 , , · · · , ; s s s

(2.285)

,

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 69     Corollary 2.24. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 1 B (α, 1)



1 −1

  (1 − t)β−1 p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt

= 2β × p+s Supersinq+s

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s β+2 β+s c1 , · · · , cq , 1+β s , s ,··· , s ;

(2.286)

2s λ ,

where λ is a constant.     Corollary 2.25. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, |z| < 1,  q p Re k=1 ck − k=1 ak > 0, and κ ∈ N, then 

1 B (α, 1) = zα

z

  t α−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt

0



× p+κ Supercos q+κ

a1 , · · · , ap , κ1 , κ2 , · · · , 1; λzκ α+2 α+κ c1 , · · · , cq , α+1 , , · · · , ; κ κ κ

(2.287)

,

where λ is a constant.   Corollary 2.26. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , q    p Re cq > 0, |z| < 1, Re k=1 ck − k=1 ak > 0, and κ ∈ N, then 1 B (1, β)



z

  (z − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt

0

= zβ



× p+κ Supercos q+κ

a1 , · · · , ap , κ1 , κ2 , · · · , 1; κ 2+β β+κ λz , , · · · , ; c1 , · · · , cq , 1+β κ κ κ

(2.288)

,

where λ is a constant.     Corollary 2.27. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0,  q p Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

  t α−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; α λz , = z × p+1 Supercos q+1 c1 , · · · , cq , α + 1;

1 B (α, 1)

where λ is a constant.

z

(2.289)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

    Corollary 2.28. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, q  p Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

  (z − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , 1; β λz , = z × p+1 Supercos q+1 c1 , · · · , cq , 1 + β;

1 B (1, β)

z

(2.290)

where λ is a constant.     Corollary 2.29. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and q  p Re k=1 ck − k=1 ak > 0, then 

  t α−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; λ , = p+1 Supercos q+1 c1 , · · · , cq , α + 1;

1 B (α, 1)

1

(2.291)

where λ is a constant.     Corollary 2.30. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and q  p Re k=1 ck − k=1 ak > 0, then 

  (1 − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , 1; λ , = p+1 Supercos q+1 c1 , · · · , cq , 1 + β;

1 B (1, β)

1

(2.292)

where λ is a constant.   Corollary 2.31. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have 

z

  t α−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt

0

= B (α, 1) zα × p+κ+s Supercos q+κ+s where λ is a constant.



, 1s , · · · , 1; κ κ s s λzκ+s a1 , · · · , ap , ακ , · · · , α+κ−1 κ , α+κ+s (κ + s)κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

(2.293)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 71     Corollary 2.32. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have  z   (z − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt 0

= B (1, β) zβ



× p+κ+s Supercos q+κ+s

; a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 s 1+β β+κ+s c1 , · · · , cq , κ+s , · · · , κ+s ;

κ κ s s λzκ+s

(2.294)

,

(κ + s)κ+s

where λ is a constant.

    Corollary 2.33. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   t α−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt 0

= B (α, 1)



× p+κ+s Supercos q+κ+s

(2.295)



, 1s , · · · , 1; a1 , · · · , ap , ακ , · · · , α+κ−1 κ

κ κ ss λ

α+κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

(κ + s)κ+s

,

where λ is a constant.

    Corollary 2.34. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   (1 − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt 0

= B (1, β)



× p+κ+s Supercos q+κ+s

; a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 s 1+β β+κ+s c1 , · · · , cq , κ+s , · · · , κ+s ;

(2.296)



κ κ ss λ

,

(κ + s)κ+s

where λ is a constant.

    Corollary 2.35. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   1 (1 + t)α−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt B (α, 1) −1 = 2α × p+κ+s Supercos q+κ+s where λ is a constant.



, 1s , · · · , ss ; a1 , · · · , ap , ακ , · · · , α+κ−1 κ α+κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

κ κ s s λ2κ+s (κ + s)κ+s

(2.297)

,

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

    Corollary 2.36. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have 1 B (1, β)



1

−1

  (1 − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt

= 2β



κ s s λ2κ+s ; a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 κ s × p+κ+s Supercos q+κ+s , β+κ+s (κ + s)κ+s c1 , · · · , cq , 1+β , · · · , ; κ+s κ+s

(2.298)

where λ is a constant.     Corollary 2.37. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have  z   1 t α−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt B (α, 1) 0 = zα



× p+s Supercos q+s



a1 , · · · , ap , 1s , 2s , · · · , 1; α+2 α+s c1 , · · · , cq , α+1 s , s ,··· , s ;

(2.299)

λzs ,

where λ is a constant.     Corollary 2.38. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have  z   1 (z − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt B (1, β) 0 = zβ × p+s Supercos q+s

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s 2+β β+s c1 , · · · , cq , 1+β s , s ,··· , s ;



(2.300)

λzs ,

where λ is a constant.     Corollary 2.39. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  t α−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

a1 , · · · , ap , 1s , 2s , · · · , 1; λ , = p+s Supercos q+s α+2 α+s c1 , · · · , cq , α+1 s , s ,··· , s ;

1 B (α, 1)

where λ is a constant.

1

(2.301)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 73     Corollary 2.40. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  (1 − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s = p+s Supercos q+s λ , β+2 β+s c1 , · · · , cq , 1+β s , s ,··· , s ;

1 B (1, β)

1

(2.302)

where λ is a constant.     Corollary 2.41. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 1 B (α, 1) =2



1

−1

  (1 + t)α−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt

α



× p+s Supercos q+s

a1 , · · · , ap , 1s , 2s , · · · , 1; 2s λ α+2 α+s c1 , · · · , cq , α+1 , , · · · , ; s s s

(2.303)

,

where λ is a constant.     Corollary 2.42. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 1 B (α, 1)



1 −1

  (1 − t)β−1 p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt

= 2β × p+s Supercos q+s



a1 , · · · c1 , · · ·

β+s−1 , ap , βs , β+1 ; s s ,··· , s 2λ 1+β β+2 β+s , cq , s , s , · · · , s ;

(2.304)

,

where λ is a constant.

2.3 The hypergeometric superhyperbolic functions via Clausen hypergeometric series In this section, comparing with the classical hyperbolic functions for the exponential function, we now proceed to apply the idea to set up the hypergeometric superhyperbolic functions for the Clausen hypergeometric series. We discuss in detail the Laplace transforms for the hypergeometric superhyperbolic functions via Clausen hypergeometric series.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

2.3.1 Definitions, convergence, properties, and theorems for the hypergeometric superhyperbolic functions via Clausen hypergeometric series We now present the definitions of the hypergeometric superhyperbolic functions for the Clausen hypergeometric series. Definition 2.9. The hypergeometric superhyperbolic sine for the Clausen hypergeometric series is defined as   ∞ (a1 )2n+1 · · · ap 2n+1 z2n+1      , (2.305) p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; z = (c1 )2n+1 · · · cq 2n+1 (2n + 1)! n=0

where an , cn ∈ C and n, p, q ∈ N0 . The convergence for the hypergeometric superhyperbolic sine of the Clausen hypergeometric series is presented as follows: 1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, p q it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0. Definition 2.10. The hypergeometric superhyperbolic cosine for the Clausen hypergeometric series is defined as   ∞  (a1 )2n · · · ap 2n z2n     , (2.306) p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; z = (c1 )2n · · · cq 2n (2n)! n=0

where an , cn ∈ C and n, p, q ∈ N0 . The cases of convergence for the hypergeometric superhyperbolic cosine of the Clausen hypergeometric series are presented as follows: 1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, p q it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0. Definition 2.11. The hypergeometric superhyperbolic tangent for the Clausen hypergeometric series is defined as     p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; z   , (2.307) p Supertanhq a1 , · · · , ap ; c1 , · · · , cq ; z = p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; z where an , cn ∈ C and n, p, q ∈ N0 . The cases of convergence for the hypergeometric superhyperbolic tangent of the Clausen hypergeometric series is presented as follows:

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 75

1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, q p it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0. Definition 2.12. The hypergeometric superhyperbolic cotangent for the Clausen hypergeometric series is defined as     p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; z   , (2.308) p Supercothq a1 , · · · , ap ; c1 , · · · , cq ; z = p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; z where an , cn ∈ C and n, p, q ∈ N0 . The cases of convergence for the hypergeometric superhyperbolic cotangent of the Clausen hypergeometric series are presented as follows: 1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, q p it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0. Definition 2.13. The hypergeometric superhyperbolic secant for the Clausen hypergeometric series is defined as   1   , (2.309) p Supersechq a1 , · · · , ap ; c1 , · · · , cq ; z = p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; z where an , cn ∈ C and n, p, q ∈ N0 . The cases of convergence for the hypergeometric superhyperbolic secant of the Clausen hypergeometric series are presented as follows: 1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, q p it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0. Definition 2.14. The hypergeometric superhyperbolic cosecant for the Clausen hypergeometric series is defined as   1   , (2.310) q Supercosechq a1 , · · · , ap ; c1 , · · · , cq ; z = q Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; z where an , cn ∈ C and n, p, q ∈ N0 . The cases of convergence for the hypergeometric superhyperbolic cosecant of the Clausen hypergeometric series are presented as follows: 1. p < q: the series converges absolutely for z ∈ C; 2. p = q + 1: the series converges absolutely for |z| 1, and for |z| = 1, q p it converges absolutely for Re k=1 ck − k=1 ak > 0; 3. p > q + 1: the series converges only for z = 0.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

In the present case, we take into consideration that   p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λz     1 = p Fq a1 , · · · , ap ; c1 , · · · , cq ; λz − p Fq a1 , · · · , ap ; c1 , · · · , cq ; −λz 2   ∞  (a1 )2n+1 · · · ap 2n+1 (λz)2n+1   = , (c1 )2n+1 · · · cq 2n+1 (2n + 1)! n=0   p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λz     1 = p Fq a1 , · · · , ap ; c1 , · · · , cq ; λz + p Fq a1 , · · · , ap ; c1 , · · · , cq ; −λz 2   ∞  (a1 )2n · · · ap 2n (λz)2n   , = (c1 )2n · · · cq 2n (2n)! n=0     p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λz  , p Supertanhq a1 , · · · , ap ; c1 , · · · , cq ; λz = p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λz     p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λz  , p Supercothq a1 , · · · , ap ; c1 , · · · , cq ; λz = p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λz p Supersechq

  a1 , · · · , ap ; c1 , · · · , cq ; λz =

(2.311)

(2.312)

(2.313) (2.314)

1   , (2.315) p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λz

and q Supercosechq

  a1 , · · · , ap ; c1 , · · · , cq ; λz =

1 ,  Supersinh , · · · , ap ; c1 , · · · , cq ; λz a q q 1 (2.316)

where an , cn , λ ∈ C and n, p, q ∈ N0 . Definition 2.15. The set of the hypergeometric superhyperbolic functions including   the hypergeometric superhyperbolic sine p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; z , hyper  geometric superhyperbolic cosine p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; z , hypergeometric   superhyperbolic tangent p Supertanhq a1 , · · · , ap ; c1 , · · · , cq ; z , hypergeometric superhy  perbolic cotangent p Supercothq a1 , · · · , ap ; c1 , · · · , cq ; z , hypergeometric superhyperbolic   secant p Supersechq a1 , · · · , ap ; c1 , · · · , cq ; z , and hypergeometric superhyperbolic cose  cant p Supercosechq a1 , · · · , ap ; c1 , · · · , cq ; z is called the hypergeometric superhyperbolic functions. By the concepts for the hypergeometric superhyperbolic functions, we find the following property. Property 2.5. Suppose that an , cn ∈ C and n, p, q ∈ N0 . Then we have p Supersinhq

    a1 , · · · , ap ; c1 , · · · , cq ; −z = −p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; z , (2.317)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 77     a1 , · · · , ap ; c1 , · · · , cq ; −z = p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; z ,     p Supertanhq a1 , · · · , ap ; c1 , · · · , cq ; −z = −p Supertanhq a1 , · · · , ap ; c1 , · · · , cq ; z ,     p Supercothq a1 , · · · , ap ; c1 , · · · , cq ; −z = −p Supercothq a1 , · · · , ap ; c1 , · · · , cq ; z ,     p Supersechq a1 , · · · , ap ; c1 , · · · , cq ; −z = p Supersechq a1 , · · · , ap ; c1 , · · · , cq ; z , p Supercoshq

(2.318) (2.319) (2.320) (2.321)

and p Supercosechq

    a1 , · · · , ap ; c1 , · · · , cq ; −z = −p Supercosechq a1 , · · · , ap ; c1 , · · · , cq ; z . (2.322)

Using these equations in the properties, we further conclude the following result. Property 2.6. Suppose that an , cn ∈ C and n, p, q ∈ N0 . Then we have     p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; z = −i p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; iz , (2.323)     p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; −z = p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; iz , (2.324)     p Supertanhq a1 , · · · , ap ; c1 , · · · , cq ; −z = −i p Supertanq a1 , · · · , ap ; c1 , · · · , cq ; iz , (2.325)     p Supercothq a1 , · · · , ap ; c1 , · · · , cq ; −z = i p Supercot q a1 , · · · , ap ; c1 , · · · , cq ; iz , (2.326)     (2.327) p Supersechq a1 , · · · , ap ; c1 , · · · , cq ; −z = p Supersec q a1 , · · · , ap ; c1 , · · · , cq ; iz , and     a1 , · · · , ap ; c1 , · · · , cq ; −z = i p Supercosec q a1 , · · · , ap ; c1 , · · · , cq ; iz . (2.328) Let us first evaluate   p Fq a1 , · · · , ap ; c1 , · · · , cq ; λz   (2.329) = p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λz   + p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; −λz , p Supercosechq

where an , cn , λ ∈ C and n, p, q ∈ N0 . Theorem 2.104. If p ≤ q + 1, Re (c1 ) > Re (a1 ) > 0, an , cn , λ ∈ C, n, p, q ∈ N0 , and |z| < 1, then we have   p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λz  1 (2.330)   t a1 −1 (1 − t)c1 −a1 −1 p−1 Supersinhq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt, = 0

where

  a2 , · · · , ap ; c1 , · · · , cq ; λzt     1 = p−1 Fq−1 a2 , · · · , ap ; c1 , · · · , cq ; λz − p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; −λz . 2

p−1 Supersinhq−1

(2.331)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Proof. From p Fq



  a1 , · · · , ap ; c1 , · · · , cq ; λz

=

1

  t a1 −1 (1 − t)c1 −a1 −1 p−1 Fq−1 a1 , · · · , ap ; c2 , · · · , cq ; λzt dt

(2.332)

0

and p Fq



  a1 , · · · , ap ; c1 , · · · , cq ; −λz

=

1

  t a1 −1 (1 − t)c1 −a1 −1 p−1 Fq−1 a1 , · · · , ap ; c2 , · · · , cq ; −λzt dt

(2.333)

0

it follows that   a1 , · · · , ap ; c1 , · · · , cq ; λz     1  = p Fq a1 , · · · , ap ; c1 , · · · , cq ; λz − p Fq a1 , · · · , ap ; c1 , · · · , cq ; −λz 2i  1   = t a1 −1 (1 − t)c1 −a1 −1 p−1 Supersinhq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt

p Supersinhq

(2.334)

0

since   a2 , · · · , ap ; c1 , · · · , cq ; λzt     1 = p−1 Fq−1 a2 , · · · , ap ; c1 , · · · , cq ; λz − p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; −λz 2

p−1 Supersinhq−1

(2.335)

for p ≤ q + 1, Re (c1 ) > Re (a1 ) > 0, an , cn , λ ∈ C, and n, p, q ∈ N0 . Theorem 2.105. If p ≤ q + 1, Re (c1 ) > Re (a1 ) > 0, an , cn , λ ∈ C, n, p, q ∈ N0 , and |z| < 1, then we have   p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λz  1 (2.336)   t a1 −1 (1 − t)c1 −a1 −1 p−1 Supercoshq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt, = 0

where   a2 , · · · , ap ; c1 , · · · , cq ; λzt     1 = p−1 Fq−1 a2 , · · · , ap ; c1 , · · · , cq ; λz + p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; −λz . 2

p−1 Supercoshq−1

Proof. By the equalities   p Fq a1 , · · · , ap ; c1 , · · · , cq ; λz  1   t a1 −1 (1 − t)c1 −a1 −1 p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt = 0

(2.337)

(2.338)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 79

and p Fq



  a1 , · · · , ap ; c1 , · · · , cq ; −λz

=

1

  t a1 −1 (1 − t)c1 −a1 −1 p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; −λzt dt

(2.339)

0

we may write   a1 , · · · , ap ; c1 , · · · , cq ; λz     1 = p Fq a1 , · · · , ap ; c1 , · · · , cq ; λz + p Fq a1 , · · · , ap ; c1 , · · · , cq ; −λz 2  1   t a1 −1 (1 − t)c1 −a1 −1 p−1 Supercoshq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt, =

p Supercoshq

(2.340)

0

since

  a2 , · · · , ap ; c1 , · · · , cq ; λzt     1 = p−1 Fq−1 a2 , · · · , ap ; c1 , · · · , cq ; λz + p−1 Fq−1 a2 , · · · , ap ; c2 , · · · , cq ; −λz 2

p−1 Supercoshq−1

(2.341)

for Re (c1 ) > Re (a1 ) > 0, an , cn , λ ∈ C, and n, p, q ∈ N0 . √ Property 2.7. Suppose that Re (c1 ) > Re (a1 ) > 0, i = −1, an , cn , μ, λ, z ∈ C, and n, p, q ∈ N0 . The hypergeometric series of the form μp Fq ((a) , (c) ; iλz)   = μp Fq a1 , · · · , ap ; c1 , · · · , cq ; iλz   a1 , · · · , ap = μ p Fq ; iλz c1 , · · · , cq   ∞  (a1 )n · · · ap n (iλz)n   =μ (c1 )n · · · cq n n! n=0   = μp Supercos q a1 , · · · , ap ; c1 , · · · , cq ; λz   + μi p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; λz ,

(2.342)

is a solution of the differential equation (Q (q, μ, λ, cn ) ϕ)2 (z) − (P (p, μ, λ, an ) ϕ)2 (z) = 0,

(2.343)

where (Q (q, μ, λ, cn ) ϕ)2 (z)  q  d  d ((Q (q, μ, λ, cn ) ϕ) (z)) z + (cn − 1) ((Q (q, μ, λ, cn ) ϕ) (z)) = μq z dz dz n=1

and

(2.344)

80

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

(P (p, μ, λ, an ) ϕ)2 (z)  p   d ((P (p, μ, λ, an ) ϕ) (z)) z + an ((P (p, μ, λ, an ) ϕ) (z)) = μp λz dz

(2.345)

n=1

with (Q (q, μ, λ, cn ) ϕ) (z)  q   dϕ (z) q d z + (cn − 1) ϕ (z) =μ z dz dz n=1 q  

 d q d z + (cn − 1) ϕ (z) =μ z dz dz

(2.346)

n=1

and (P (p, μ, λ, an ) ϕ) (z)  p   dϕ (z) z + an ϕ (z) = μp λz dz n=1    p  d p = μ λz z + an ϕ (z) . dz

(2.347)

n=1

Proof. By means of the equation   ∞    (a1 )n · · · ap n (iλz)n   μp Fq a1 , · · · , ap ; c1 , · · · , cq ; iλz = μ (c1 )n · · · cq n n! n=0

(2.348)

we take into consideration that (Q (q, μ, λ, cn ) ϕ) (z) − i (P (p, μ, λ, an ) ϕ) (z) = 0.

(2.349)

By the equality   ∞    (a1 )n · · · ap n (−iλz)n   μp Fq a1 , · · · , ap ; c1 , · · · , cq ; −iλz = μ n! (c1 )n · · · cq n n=0

(2.350)

we further conclude that (Q (q, μ, λ, cn ) ϕ) (z) + i (P (p, μ, λ, an ) ϕ) (z) = 0.

(2.351)

Substituting (2.349) into (2.351), we have that (Q (q, μ, λ, cn ) ϕ)2 (z) + (P (p, μ, λ, an ) ϕ)2 (z) = 0, √ where Re (c1 ) > Re (a1 ) > 0, i = −1, an , cn , μ, λ, z ∈ C, and n, p, q ∈ N0 . Therefore we finish the proof.

(2.352)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 81

2.3.2 Laplace transforms for the hypergeometric superhyperbolic functions via Clausen hypergeometric series In this section, we now consider the Laplace transforms for the hypergeometric superhyperbolic functions via Clausen hypergeometric series. Theorem 2.106. Let an , cn , z, λ ∈ C and n, p, q ∈ N0 . Then the Laplace transforms of the function t λ−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; zt is as follows:    L t λ−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; zt (2.353) z  (λ) , = λ p+1 Supersinhq λ, a1 , · · · , ap ; c1 , · · · , cq ; s s where the convergence cases are as follows: 1. Re (λ) > 0, p ≤ q; 2. Re (s) > 0 if p < q; 3. Re (s) > Re (z) if p = q. Proof. Using the integral representations    L t λ−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; zt     ∞ a , · · · , ap = e−st t λ−1 p Fq 1 ; zt dt c1 , · · · , cq 0    (λ) λ, a1 , · · · , ap z = λ p+1 Fq ; c1 , · · · , cq s s and

   L t λ−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; −zt     ∞ a 1 , · · · , ap −st λ−1 t = e ; −zt dt p Fq c1 , · · · , cq 0    (λ) λ, a1 , · · · , ap z = λ p+1 Fq ;− , c1 , · · · , cq s s

(2.354)

(2.355)

we have the result. Theorem 2.107. Let an , cn , z, λ ∈ C and n, p, q ∈ N0 .  λ−1 Supersinh a , · · · , a ; c , · · · , c ; q 1 p 1 q p  Then there the Laplace transform of the function t zt is as follows:    L t λ−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; zt (2.356) z  (λ) , = λ p+1 Supercoshq λ, a1 , · · · , ap ; c1 , · · · , cq ; s s

82

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

where the convergence cases are as follows: 1. Re (λ) > 0 if p ≤ q; 2. Re (s) > 0 if p < q; 3. Re (s) > Re (z) if p = q. Proof. In view of the facts that    L t λ−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; zt     ∞ a , · · · , ap = e−st t λ−1 p Fq 1 ; zt dt c1 , · · · , cq 0    (λ) λ, a1 , · · · , ap z = λ p+1 Fq ; c1 , · · · , cq s s and

   L t λ−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; −zt     ∞ a , · · · , ap = e−st t λ−1 p Fq 1 ; −zt dt c1 , · · · , cq 0    (λ) λ, a1 , · · · , ap z = λ p+1 Fq ;− , c1 , · · · , cq s s

(2.357)

(2.358)

we get the result. For the moment, let us consider the following theorems without proofs.     Theorem If p ≤ q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, q 2.108. p Re k=1 ck − k=1 ak > 0, and |z| < 1, then   p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λz  (c1 ) = (2.359)  (a1 )  (c1 − a1 )  1   t a1 −1 (1 − t)c1 −a1 −1 p−1 Supersinhq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt, × 0

where λ is a constant.     Theorem If p ≤ q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, q 2.109. p Re k=1 ck − k=1 ak > 0, and |z| < 1, then we have   a1 , · · · , ap , a; λz p+1 Supersinhq+1 c1 , · · · , cq , c;  (c) (2.360) =  (a)  (c − a)  1   × t a−1 (1 − t)c−a−1 p Supersinhq a1 , a2 , · · · , ap ; c1 , c2 , · · · , cq ; λzt dt, 0

where λ is a constant.

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 83   Theorem 2.110. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , q    p Re cq > 0, |z| < 1, Re k=1 ck − k=1 ak > 0, and κ ∈ N, then 1 B (α, β) =z



z

  t α−1 (z − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt

0

α+β−1



× p+κ Supersinhq+κ

α+κ−1 ; a1 , · · · , ap , ακ , α+1 κ ,··· , κ λzκ α+β α+β+1 α+β+κ−1 ; c1 , · · · , cq , κ , κ , · · · , κ

(2.361)

,

where λ is a constant.   Theorem 2.111. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , q    p Re cq > 0, Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

  t α−1 (z − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; α+β−1 λz , =z × p+1 Supersinhq+1 c1 , · · · , cq , α + β;

1 B (α, β)

z

(2.362)

where λ is a constant.   Theorem 2.112. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , q    p Re cq > 0, and Re k=1 ck − k=1 ak > 0, then 

  t α−1 (1 − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; λ , = p+1 Supersinhq+1 c1 , · · · , cq , α + β;

1 B (α, β)

1

(2.363)

where λ is a constant.   Theorem 2.113. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have 

z

  t α−1 (z − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt

0

= B (α, β) zα+β−1 × p+κ+s Supersinhq+κ+s where λ is a constant.



, βs , · · · , β+s−1 ; a1 , · · · , ap , ακ , · · · , α+κ−1 κ s α+β α+β+κ+s−1 ; c1 , · · · , cq , κ+s , · · · , κ+s

κ κ s s λzκ+s (κ + s)κ+s

,

(2.364)

84

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

  Theorem 2.114. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have 

1

  t α−1 (1 − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt

0

= B (α, β)



× p+κ+s Supersinhq+κ+s



, βs , · · · , β+s−1 ; a1 , · · · , ap , ακ , · · · , α+κ−1 κ s

κ κ ss λ

α+β+κ+s−1 ; c1 , · · · , cq , α+β κ+s , · · · , κ+s

(κ + s)κ+s

(2.365)

,

where λ is a constant.

  Theorem 2.115. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have 1 B (α, β)



1

−1

  (1 + t)α−1 (1 − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt

= 2α+β−1



× p+κ+s Supersinhq+κ+s

, βs , · · · , β+s−1 ; κ κ s s λ2κ+s a1 , · · · , ap , ακ , · · · , α+κ−1 κ s α+β α+β+κ+s−1 (κ + s)κ+s c1 , · · · , cq , κ+s , · · · , ; κ+s

, (2.366)

where λ is a constant.

  Theorem 2.116. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have  z   1 t α−1 (z − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt B (α, β) 0 = zα+β−1



× p+s Supersinhq+s

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s α+β+1 c1 , · · · , cq , α+β , · · · , α+β+s−1 ; s , s s



(2.367)

λzs ,

where λ is a constant.

  Theorem 2.117. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  t α−1 (1 − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

β+s−1 a1 , · · · , ap , βs , β+1 , · · · , ; s s = p+s Supersinhq+s λ , α+β+1 α+β+s−1 c1 , · · · , cq , α+β , , · · · , ; s s s

1 B (α, β)

1

where λ is a constant.

(2.368)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 85   Theorem 2.118. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 1 B (α, β)



1 −1

  (1 + t)α−1 (1 − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt

= 2α+β−1



× p+s Supersinhq+s

(2.369)



β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s 2s λ α+β α+β+1 α+β+s−1 c1 , · · · , cq , s , s , · · · , ; s

,

where λ is a constant.

    Theorem If p ≤ q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, q 2.119. p Re k=1 ck − k=1 ak > 0, and |z| < 1, then   p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λz  (c1 ) = (2.370)  (a1 )  (c1 − a1 )  1   t a1 −1 (1 − t)c1 −a1 −1 p−1 Supercoshq−1 a2 , · · · , ap ; c2 , · · · , cq ; λzt dt, × 0

where λ is a constant.

    Theorem If p ≤ q + 1, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, q 2.120. p Re k=1 ck − k=1 ak > 0, and |z| < 1, then we have p+1 Supercoshq+1

  a1 , · · · , ap , a; λz c1 , · · · , cq , c;

 (c)  (a)  (c − a)  1   × t a−1 (1 − t)c−a−1 p Supercoshq a1 , a2 , · · · , ap ; c1 , c2 , · · · , cq ; λzt dt, =

(2.371)

0

where λ is a constant.

  Theorem 2.121. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,     q p Re cq > 0, |z| < 1, Re k=1 ck − k=1 ak > 0, and κ ∈ N, then  z   1 t α−1 (z − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt B (α, β) 0 = zα+β−1 × p+κ Supercoshq+κ where λ is a constant.



α+κ−1 ; a1 , · · · , ap , ακ , α+1 κ ,··· , κ κ α+β α+β+1 α+β+κ−1 λz ; c1 , · · · , cq , κ , κ , · · · , κ

(2.372)

,

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

  Theorem 2.122. qIf Re (α)>p 0, Re  (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, Re k=1 ck − k=1 ak > 0, and |z| < 1, then  z   1 t α−1 (z − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt dt B (α, β) 0   (2.373) a1 , · · · , ap , α; λz , = zα+β−1 × p+1 Supercoshq+1 c1 , · · · , cq , α + β; where λ is a constant.   Theorem 2.123. If Re (α) > 0, Re (β)> 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   q p Re cq > 0, and Re k=1 ck − k=1 ak > 0, then 

  t α−1 (1 − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; λ , = p+1 Supercoshq+1 c1 , · · · , cq , α + β;

1 B (α, β)

1

(2.374)

where λ is a constant.   Theorem 2.124. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have  z   t α−1 (z − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt 0

= B (α, β) zα+β−1



, βs , · · · , β+s−1 ; κ κ s s λzκ+s a1 , · · · , ap , ακ , · · · , α+κ−1 κ s × p+κ+s Supercoshq+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c1 , · · · , cq , α+β κ+s , · · · , κ+s

(2.375)

where λ is a constant.   Theorem 2.125. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have 

1

  t α−1 (1 − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt

0

= B (α, β) × p+κ+s Supercoshq+κ+s

, βs , · · · , β+s−1 ; a1 , · · · , ap , ακ , · · · , α+κ−1 κ s α+β+κ+s−1 ; c1 , · · · , cq , α+β κ+s , · · · , κ+s

κ κ ss λ (κ + s)κ+s

(2.376)

,

where λ is a constant.   Theorem 2.126. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 87

1 B (α, β)



  (1 + t)α−1 (1 − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt

1 −1

= 2α+β−1



, βs , · · · , β+s−1 ; κ κ s s λ2κ+s a1 , · · · , ap , ακ , · · · , α+κ−1 κ s × p+κ+s Supercoshq+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c1 , · · · , cq , α+β κ+s , · · · , κ+s (2.377) where λ is a constant.   Theorem 2.127. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have  z   1 t α−1 (z − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt B (α, β) 0 = zα+β−1



× p+s Supercoshq+s

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s

α+β+1 , · · · , α+β+s−1 ; c1 , · · · , cq , α+β s , s s



(2.378)

λzs ,

where λ is a constant.   Theorem 2.128. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  t α−1 (1 − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s = p+s Supercoshq+s λ , α+β+1 , · · · , α+β+s−1 ; c1 , · · · , cq , α+β s , s s

1 B (α, β)

1

(2.379)

where λ is a constant.   Theorem 2.129. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 1 B (α, β)



1

−1

  (1 + t)α−1 (1 − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt

= 2α+β−1 × p+s Supercoshq+s where λ is a constant.



β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s 2s λ α+β α+β+1 α+β+s−1 ; c1 , · · · , cq , s , s , · · · , s

,

(2.380)

88

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

    Corollary 2.43. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, |z| < 1,  q p Re k=1 ck − k=1 ak > 0, and κ ∈ N, then 

1 B (α, 1) = zα

z

  t α−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt

0



× p+κ Supersinhq+κ

a1 , · · · , ap , κ1 , κ2 , · · · , 1; λzκ α+2 α+κ c1 , · · · , cq , α+1 , , · · · , ; κ κ κ

(2.381)

,

where λ is a constant.   Corollary 2.44. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , q    p Re cq > 0, |z| < 1, Re k=1 ck − k=1 ak > 0, and κ ∈ N, then 1 B (1, β)



z

  (z − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt

0

= zβ



× p+κ Supersinhq+κ

a1 , · · · , ap , κ1 , κ2 , · · · , 1; c1 , · · ·

2+β , cq , 1+β κ , κ ,···

,

β+κ κ ;



(2.382)

λzκ ,

where λ is a constant.     Corollary 2.45. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0,  q p Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

  t α−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; λz , = zα × p+1 Supersinhq+1 c1 , · · · , cq , α + 1;

1 B (α, 1)

z

(2.383)

where λ is a constant.     Corollary 2.46. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0,  q p Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

  (z − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , 1; λz , = zβ × p+1 Supersinhq+1 c1 , · · · , cq , 1 + β;

1 B (1, β)

where λ is a constant.

z

(2.384)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 89     Corollary 2.47. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and  q p Re k=1 ck − k=1 ak > 0, then 

  t α−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; λ , = p+1 Supersinhq+1 c1 , · · · , cq , α + 1;

1 B (α, 1)

1

(2.385)

where λ is a constant.     Corollary 2.48. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and  q p Re k=1 ck − k=1 ak > 0, then 

  (1 − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , 1; λ , = p+1 Supersinhq+1 c1 , · · · , cq , 1 + β;

1 B (1, β)

1

(2.386)

where λ is a constant.   Corollary 2.49. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have 

z

  t α−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt

0

= B (α, 1) zα × p+κ+s Supersinhq+κ+s



, 1s , · · · a1 , · · · , ap , ακ , · · · , α+κ−1 κ α+κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

, 1;

κ κ s s λzκ+s

(2.387)

,

(κ + s)κ+s

where λ is a constant.     Corollary 2.50. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have 

z

  (z − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt

0

= B (1, β) zβ × p+κ+s Supersinhq+κ+s where λ is a constant.



; a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 s 1+β β+κ+s c1 , · · · , cq , κ+s , · · · , κ+s ;

κ κ s s λzκ+s (κ + s)κ+s

(2.388)

,

90

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

    Corollary 2.51. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   t α−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt 0

= B (α, 1)



× p+κ+s Supersinhq+κ+s

, 1s , · · · a1 , · · · , ap , ακ , · · · , α+κ−1 κ α+κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

, 1;

(2.389)



κ κ ss λ

,

(κ + s)κ+s

where λ is a constant.

    Corollary 2.52. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   (1 − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt 0

= B (1, β)



× p+κ+s Supersinhq+κ+s

(2.390)



; a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 s

κ κ ss λ

β+κ+s c1 , · · · , cq , 1+β κ+s , · · · , κ+s ;

(κ + s)κ+s

,

where λ is a constant.

    Corollary 2.53. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   1 (1 + t)α−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt B (α, 1) −1 = 2α × p+κ+s Supersinhq+κ+s



, 1s , · · · , ss ; a1 , · · · , ap , ακ , · · · , α+κ−1 κ α+κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

κ κ s s λ2κ+s (κ + s)κ+s

(2.391)

,

where λ is a constant.

    Corollary 2.54. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   1 (1 − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt B (1, β) −1 = 2β



κ s s λ2κ+s ; a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 κ s × p+κ+s Supersinhq+κ+s , β+κ+s (κ + s)κ+s , · · · , ; c1 , · · · , cq , 1+β κ+s κ+s where λ is a constant.

(2.392)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 91     Corollary 2.55. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have 1 B (α, 1) = zα



z

  t α−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt

0



× p+s Supersinhq+s

a1 , · · · , ap , 1s , 2s , · · · , 1; λzs α+1 α+2 α+s c1 , · · · , cq , s , s , · · · , s ;



(2.393)

,

where λ is a constant.     Corollary 2.56. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have 1 B (1, β) =z



z

  (z − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt

0

β

× p+s Supersinhq+s

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s 2+β β+s c1 , · · · , cq , 1+β s , s ,··· , s ;



(2.394)

λzs ,

where λ is a constant.     Corollary 2.57. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  t α−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

a1 , · · · , ap , 1s , 2s , · · · , 1; λ , = p+s Supersinhq+s α+2 α+s c1 , · · · , cq , α+1 s , s ,··· , s ;

1 B (α, 1)

1

(2.395)

where λ is a constant.     Corollary 2.58. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  (1 − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s = p+s Supersinhq+s λ , β+2 β+s c1 , · · · , cq , 1+β s , s ,··· , s ;

1 B (1, β)

1

where λ is a constant.

(2.396)

92

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

    Corollary 2.59. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have  1   1 (1 + t)α−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt B (α, 1) −1 = 2α



× p+s Supersinhq+s

a1 , · · · , ap , 1s , 2s , · · · , 1; 2s λ α+2 α+s c1 , · · · , cq , α+1 , , · · · , ; s s s

(2.397)

,

where λ is a constant.

    Corollary 2.60. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have  1   1 (1 − t)β−1 p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt B (α, 1) −1 = 2β



× p+s Supersinhq+s

a1 , · · · c1 , · · ·

β+s−1 , ap , βs , β+1 ; s s ,··· , s 2λ 1+β β+2 β+s , cq , s , s , · · · , s ;

(2.398)

,

where λ is a constant.

    > 0, Re > 0, · · · , Re a > 0, · · · , Re cq > 0, |z| < 1, Corollary 2.61. If Re > 0, Re (c ) ) (α) (a 1 p 1 q  p Re k=1 ck − k=1 ak > 0, and κ ∈ N, then  z   1 t α−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt B (α, 1) 0 = zα



× p+κ Supercoshq+κ

a1 , · · · , ap , κ1 , κ2 , · · · , 1; α+2 α+κ c1 , · · · , cq , α+1 κ , κ ,··· , κ ;



(2.399)

λzκ ,

where λ is a constant.

  Corollary 2.62. If Re(α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   q p Re cq > 0, |z| < 1, Re k=1 ck − k=1 ak > 0, and κ ∈ N, then  z   1 (z − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt κ dt B (1, β) 0 = zβ × p+κ Supercoshq+κ where λ is a constant.



a1 , · · · , ap , κ1 , κ2 , · · · , 1; κ 2+β β+κ λz , , · · · , ; c1 , · · · , cq , 1+β κ κ κ

(2.400)

,

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 93     Corollary 2.63. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0,  q p Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

  t α−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; λz , = zα × p+1 Supercoshq+1 c1 , · · · , cq , α + 1;

1 B (α, 1)

z

(2.401)

where λ is a constant.     Corollary 2.64. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0,  q p Re k=1 ck − k=1 ak > 0, and |z| < 1, then 

  (z − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , 1; λz , = zβ × p+1 Supercoshq+1 c1 , · · · , cq , 1 + β;

1 B (1, β)

z

(2.402)

where λ is a constant.     Corollary 2.65. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and  q p Re k=1 ck − k=1 ak > 0, then 

  t α−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , α; λ , = p+1 Supercoshq+1 c1 , · · · , cq , α + 1;

1 B (α, 1)

1

(2.403)

where λ is a constant.     Corollary 2.66. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, and  q p Re k=1 ck − k=1 ak > 0, then 

  (1 − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt dt 0   a1 , · · · , ap , 1; λ , = p+1 Supercoshq+1 c1 , · · · , cq , 1 + β;

1 B (1, β)

where λ is a constant.

1

(2.404)

94

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

  Corollary 2.67. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have  z   t α−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt 0

= B (α, 1) zα



, 1s , · · · , 1; κ κ s s λzκ+s a1 , · · · , ap , ακ , · · · , α+κ−1 κ × p+κ+s Supercoshq+κ+s , α+κ+s (κ + s)κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

(2.405)

where λ is a constant.

    Corollary 2.68. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , κ, s ∈ N, and |z| < 1, then we have  z   (z − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (z − t)s dt 0

= B (1, β) zβ



; κ κ s s λzκ+s a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 s × p+κ+s Supercoshq+κ+s , β+κ+s (κ + s)κ+s c1 , · · · , cq , 1+β κ+s , · · · , κ+s ;

(2.406)

where λ is a constant.

    Corollary 2.69. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   t α−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt 0

= B (α, 1) × p+κ+s Supercoshq+κ+s

, 1s , · · · , 1; a1 , · · · , ap , ακ , · · · , α+κ−1 κ α+κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

κ κ ss λ

(2.407)

,

(κ + s)κ+s

where λ is a constant.

    Corollary 2.70. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   (1 − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λt κ (1 − t)s dt 0

= B (1, β) × p+κ+s Supercoshq+κ+s where λ is a constant.

; a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 s β+κ+s c1 , · · · , cq , 1+β κ+s , · · · , κ+s ;

κ κ ss λ (κ + s)κ+s

(2.408)

,

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 95     Corollary 2.71. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   1 (1 + t)α−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt B (α, 1) −1 = 2α



, 1s , · · · , ss ; κ κ s s λ2κ+s a1 , · · · , ap , ακ , · · · , α+κ−1 κ × p+κ+s Supercoshq+κ+s , α+κ+s (κ + s)κ+s c1 , · · · , cq , α+1 κ+s , · · · , κ+s ;

(2.409)

where λ is a constant.

    Corollary 2.72. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and κ, s ∈ N, then we have  1   1 (1 − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)κ (1 − t)s dt B (1, β) −1 = 2β



× p+κ+s Supercoshq+κ+s

; a1 , · · · , ap , κ1 , · · · , 1, βs , · · · , β+s−1 s 1+β β+κ+s c1 , · · · , cq , κ+s , · · · , κ+s ;

κ κ s s λ2κ+s (κ + s)κ+s

(2.410)

,

where λ is a constant.

    Corollary 2.73. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have  z   1 t α−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt B (α, 1) 0 = zα



× p+s Supercoshq+s



a1 , · · · , ap , 1s , 2s , · · · , 1;

α+2 α+s c1 , · · · , cq , α+1 s , s ,··· , s ;

(2.411)

λzs ,

where λ is a constant.

    Corollary 2.74. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have  z   1 (z − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (z − t)s dt B (1, β) 0 = zβ × p+s Supercoshq+s where λ is a constant.



a1 , · · · c1 , · · ·

β+s−1 , ap , βs , β+1 ; s s ,··· , s λz 1+β 2+β β+s , cq , s , s , · · · , s ;

(2.412)

,

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

    Corollary 2.75. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  t α−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

a1 , · · · , ap , 1s , 2s , · · · , 1; = p+s Supercoshq+s λ , α+2 α+s c1 , · · · , cq , α+1 s , s ,··· , s ;

1 B (α, 1)

1

(2.413)

where λ is a constant.     Corollary 2.76. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 

  (1 − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s = p+s Supercoshq+s λ , β+2 β+s c1 , · · · , cq , 1+β s , s ,··· , s ;

1 B (1, β)

1

(2.414)

where λ is a constant.     Corollary 2.77. If Re (α) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 1 B (α, 1) =2



1

−1

  (1 + t)α−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt

α



a1 , · · · , ap , 1s , 2s , · · · , 1; 2s λ α+2 α+s c1 , · · · , cq , α+1 , , · · · , ; s s s

× p+s Supercoshq+s

(2.415)

,

where λ is a constant.     Corollary 2.78. If Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have 1 B (α, 1) =2



1

−1

  (1 − t)β−1 p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 + t)s dt

β

× p+s Supercoshq+s where λ is a constant.



a1 , · · · c1 , · · ·

β+s−1 , ap , βs , β+1 ; s s ,··· , s 2λ 1+β β+2 β+s , cq , s , s , · · · , s ;

,

(2.416)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 97

2.4 The special functions via Clausen hypergeometric series with three numerator parameters and two denominator parameters In this section, we introduce the special functions via Clausen hypergeometric series with three numerator parameters and two denominator parameters.

2.4.1 The hypergeometric supertrigonometric functions with three numerator parameters and two denominator parameters Clausen [55] introduced the well-known Clausen hypergeometric series with three numerator parameters and two denominator parameters: 3 F2 (a1 , a2 , a3 ; c1 , c2 ; z)



= 3 F2 =

a1 , a2 , a3 ;z c1 , c2



(2.417)

∞  (a1 )n (a3 )n (a3 )n zn

(c1 )n (c2 )n

n=0

n!

,

where a1 , a2 , a3 , c1 , c2 , z ∈ C and n ∈ N0 . The convergences for the Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows: The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it converges absolutely for  q p Re k=1 ck − k=1 ak > 0. Using analogous considerations to obtain the hypergeometric supertrigonometric functions, we have the following definitions. Definition 2.16. The hypergeometric supersine via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; z) =

∞  (a1 )2n+1 (a3 )2n+1 (a3 )2n+1 (−1)n z2n+1 n=0

(c1 )2n+1 (c2 )2n+1

(2n + 1)!

,

(2.418)

where a1 , a2 , a3 , c1 , c2 , z ∈ C, n ∈ N0 , and |z| < 1. The convergences for the hypergeometric supersine via Clausen hypergeometric series with three numerator parameters and two denominator parameters are as follows: The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it converges abq  p solutely for Re k=1 ck − k=1 ak > 0. From Eq. (2.383) we have the following equations: 0 (z) =

(a1 )1 (a3 )1 (a3 )1 z , (c1 )1 (c2 )1 1!

(2.419)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

1 (z) = − 2 (z) =

(2.420)

(a1 )5 (a3 )5 (a3 )5 z5 , 5! (c1 )5 (c2 )5

(2.421)

(a1 )7 (a3 )7 (a3 )7 z7 , (c1 )7 (c2 )7 71!

(2.422)

(a1 )9 (a3 )9 (a3 )9 z9 , 9! (c1 )9 (c2 )9

(2.423)

3 (z) = − 4 (z) =

(a1 )3 (a3 )3 (a3 )3 z3 , 3! (c1 )3 (c2 )3

and 5 (z) = −

(a1 )11 (a3 )11 (a3 )11 z11 , 11! (c1 )11 (c2 )11

(2.424)

where |z| < 1. Definition 2.17. The hypergeometric supercosine via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; z) =

∞  (a1 )2n (a3 )2n (a3 )2n (−1)n z2n n=0

(c1 )2n (c2 )2n

(2n)!

,

(2.425)

where a1 , a2 , a3 , c1 , c2 , z ∈ C, n ∈ N0 , and |z| < 1. The convergences for the hypergeometric supercosine via Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows: The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it q  p converges absolutely for Re k=1 ck − k=1 ak > 0. From Eq. (2.419) we obtain the following equations: 0 (z) = 1, 1 (z) = − 2 (z) =

(a1 )4 (a3 )4 (a3 )4 z4 , 4! (c1 )4 (c2 )4

3 (z) = − 4 (z) = and

(a1 )2 (a3 )2 (a3 )2 z2 , 2! (c1 )2 (c2 )2

(a1 )6 (a3 )6 (a3 )6 z6 , 6! (c1 )6 (c2 )6

(a1 )8 (a3 )8 (a3 )8 z8 , 8! (c1 )8 (c2 )8

(2.426) (2.427) (2.428) (2.429) (2.430)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions 99

5 (z) = −

(a1 )10 (a3 )10 (a3 )10 z10 , 10! (c1 )10 (c2 )10

(2.431)

where |z| < 1. Definition 2.18. The hypergeometric supertangent via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supertan2 (a1 , a2 , a3 ; c1 , c2 ; z) =

3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; z) 3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; z)

(2.432)

,

where a1 , a2 , a3 , c1 , c2 , z ∈ C, n ∈ N0 , and |z| < 1. The convergences for the hypergeometric supertangent via Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows: The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it conq  p verges absolutely for Re > 0. c − a k k k=1 k=1 Definition 2.19. The hypergeometric supercotangent via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supercot 2 (a1 , a2 , a3 ; c1 , c2 ; z) =

3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; z) 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; z)

,

(2.433)

where a1 , a2 , a3 , c1 , c2 , z ∈ C, n ∈ N0 , and |z| < 1. The convergences for the hypergeometric supercotangent via Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows: The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it con q p verges absolutely for Re k=1 ck − k=1 ak > 0. Definition 2.20. The hypergeometric supersecant via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supersec 2 (a1 , a2 , a3 ; c1 , c2 ; z) =

1 , 3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; z)

(2.434)

where a1 , a2 , a3 , c1 , c2 , z ∈ C, n ∈ N0 , and |z| < 1. The convergences for the hypergeometric supersecant via Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows:

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it con q p > 0. c − a verges absolutely for Re k=1 k k=1 k Definition 2.21. The hypergeometric supersecant via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supercosec 2 (a1 , a2 , a3 ; c1 , c2 ; z) =

1 , 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; z)

(2.435)

where a1 , a2 , a3 , c1 , c2 , z ∈ C, n ∈ N0 , and |z| < 1. The convergences for the hypergeometric supersecant via Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows: |z| < 1, and diverges for |z| > 1, and for |z| = 1, it The series converges absolutely q for  p converges absolutely for Re c − k=1 ak > 0. k k=1 Using the equalities 3 F2 (a1 , a2 , a3 ; c1 , c2 ; λz)



= 3 F2 =

a1 , a2 , a3 ; λz c1 , c2



(2.436)

∞  (a1 )n (a3 )n (a3 )n λn zn

n!

(c1 )n (c2 )n

n=0

,

3 F2 (a1 , a2 , a3 ; c1 , c2 ; −λz)

= 3 F2 =

∞  n=0

  a1 , a2 , a3 ; −λz c1 , c2 (a1 )n (a3 )n (a3 )n (c1 )n (c2 )n

(2.437)

(−λ)n zn n!

,

3 F2 (a1 , a2 , a3 ; c1 , c2 ; iz)



= 3 F2 =

∞  n=0

and

a1 , a2 , a3 ; iz c1 , c2



(2.438) n

(a1 )n (a3 )n (a3 )n (iz) , n! (c1 )n (c2 )n

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

101

3 F2 (a1 , a2 , a3 ; c1 , c2 ; −iz)



= 3 F2 =

a1 , a2 , a3 ; −iz c1 , c2



(2.439)

∞  (a1 )n (a3 )n (a3 )n (−iz)n n=0

(c1 )n (c2 )n

n!

,

we find that 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; λz)

1 (3 F2 (a1 , a2 , a3 ; c1 , c2 ; iλz) − 3 F2 (a1 , a2 , a3 ; c1 , c2 ; −iλz)) 2i ∞  (a1 )2n+1 (a3 )2n+1 (a3 )2n+1 (−1)n (λz)2n+1 = , (c1 )2n+1 (c2 )2n+1 (2n + 1)!

=

(2.440)

n=0

3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; λz)

1 (3 F2 (a1 , a2 , a3 ; c1 , c2 ; iλz) + 3 F2 (a1 , a2 , a3 ; c1 , c2 ; −iλz)) 2 ∞  (a1 )2n (a3 )2n (a3 )2n (−1)n (λz)2n , = (c1 )2n (c2 )2n (2n)!

=

(2.441)

n=0

3 Supertan2 (a1 , a2 , a3 ; c1 , c2 ; λz) = 3 Supercot 2 (a1 , a2 , a3 ; c1 , c2 ; λz) = 3 Supersec 2 (a1 , a2 , a3 ; c1 , c2 ; λz) =

3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; λz)

,

(2.442)

,

(2.443)

1 , Supercos , (a 3 2 1 a2 , a3 ; c1 , c2 ; λz)

(2.444)

3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; λz) 3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; λz) 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; λz)

and 3 Supercosec 2 (a1 , a2 , a3 ; c1 , c2 ; λz) =

1 , Supersin , (a 3 2 1 a2 , a3 ; c1 , c2 ; λz)

(2.445)

where a1 , a2 , a3 , c1 , c2 , λ, z ∈ C, and n ∈ N0 . Definition 2.22. The set of the hypergeometric supertrigonometric functions with three numerator parameters and two denominator parameters including the hypergeometric  supersine a1 , a2 , a3 ; Supersin , a , a ; c , c ; z), hypergeometric supercosine Supercos (a 3 2 1 2 3 1 2 3 2  c1 , c2 ; z , hypergeometric supertangent 3 Supertan2 (a1 , a2 , a3 ; c1 , c2 ; z), hypergeometric  supercotangent a1 , a2 , Supercot , a , a ; c , c ; z), hypergeometric supersecant Supersec (a 3 2 1 2 3 1 2 3 2  a3 ; c1 , c2 ; z , and hypergeometric supercosecant 3 Supercosec 2 (a1 , a2 , a3 ; c1 , c2 ; z) is called the hypergeometric supertrigonometric functions with three numerator parameters and two denominator parameters. For the moment, we consider to the following property.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Property 2.8. If a1 , a2 , a3 , c1 , c2 , λ, z ∈ C, and n ∈ N0 , then we have 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; −z) = −3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; z) ,

(2.446)

3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; −z) = 3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; z) ,

(2.447)

3 Supertan2 (a1 , a2 , a3 ; c1 , c2 ; −z) = −3 Supertan2 (a1 , a2 , a3 ; c1 , c2 ; z),

(2.448)

3 Supercot 2 (a1 , a2 , a3 ; c1 , c2 ; −z) = −3 Supercot 2 (a1 , a2 , a3 ; c1 , c2 ; z) ,

(2.449)

3 Supersec 2 (a1 , a2 , a3 ; c1 , c2 ; −z) = 3 Supersec 2 (a1 , a2 , a3 ; c1 , c2 ; z) ,

(2.450)

3 Supercosec 2 (a1 , a2 , a3 ; c1 , c2 ; −z) = −3 Supercosec 2 (a1 , a2 , a3 ; c1 , c2 ; z).

(2.451)

and

Proof. We obtain the results directly from the definitions. By the definitions of the hypergeometric supertrigonometric functions with three numerator parameters and two denominator parameters we take into consideration that 3 F2 (a1 , a2 , a3 ; c1 , c2 ; iλz)

= 3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; λz) + i 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; λz)

(2.452)

and 3 F2 (a1 , a2 , a3 ; c1 , c2 ; λz)

= 3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; λz) + 3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; −λz) ,

(2.453)

where a1 , a2 , a3 , c1 , c2 , z ∈ C, and n ∈ N0 . Theorem 2.130. Suppose that a1 , a2 , a3 , c1 , c2 , z ∈ C and Re (c1 ) > Re (a1 ) > 0. Then we have 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; λz)



=

1

t a1 −1 (1 − t)c1 −a1 −1 2 Supersin1 (a2 , a3 ; c2 ; λzt)dt,

(2.454)

0

where 2 Supersin1 (a2 , a3 ; c2 ; λz) =

1 (2 F1 (a2 , a3 ; c2 ; iλz) − 2 F1 (a2 , a3 ; c2 ; iλz)) . 2i

(2.455)

Proof. By the equations 

1

3 F2 (a1 , a2 , a3 ; c1 , c2 ; iλz) =

t a1 −1 (1 − t)c1 −a1 −1 2 F1 (a2 , a3 ; c2 ; iλzt)dt

(2.456)

t a1 −1 (1 − t)c1 −a1 −1 2 F1 (a2 , a3 ; c2 ; −iλzt)dt

(2.457)

0

and 

1

3 F2 (a1 , a2 , a3 ; c1 , c2 ; −iλz) = 0

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

103

we can say that 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; λz)

1 (2 F1 (a1 , a2 , a3 ; c1 , c2 ; iλz) − 2 F1 (a1 , a2 , a3 ; c1 , c2 ; −iλz)) 2i  1 = t a1 −1 (1 − t)c1 −a1 −1 2 Supersin1 (a2 , a3 ; c2 ; λzt)dt

=

(2.458)

0

since 2 Supersin1 (a2 , a3 ; c1 , c2 ; λz) =

1 (2 F1 (a2 , a3 ; c1 , c2 ; iλz) − 2 F1 (a2 , a3 ; c1 , c2 ; −iλz)) (2.459) 2i

where a1 , a2 , a3 , c1 , c2 , z ∈ C and Re (c1 ) > Re (a1 ) > 0. Theorem 2.131. Suppose that a1 , a2 , a3 , c1 , c2 , z ∈ C and Re (c1 ) > Re (a1 ) > 0. Then we have 

1

3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; λz) =

t a1 −1 (1 − t)c1 −a1 −1 2 Supercos 1 (a2 , a3 ; c2 ; λzt)dt,

0

(2.460)

where 2 Supercos 1 (a2 , a3 ; c2 ; λz) =

1 (2 F1 (a2 , a3 ; c2 ; iλz) + 2 F1 (a2 , a3 ; c2 ; −iλz)) . 2

(2.461)

Proof. Putting 

1

3 F2 (a1 , a2 , a3 ; c1 , c2 ; iλz) =

t a1 −1 (1 − t)c1 −a1 −1 2 F1 (a2 , a3 ; c2 ; iλzt)dt

(2.462)

t a1 −1 (1 − t)c1 −a1 −1 2 F1 (a2 , a3 ; c2 ; −iλzt)dt,

(2.463)

0

and  3 F2 (a1 , a2 , a3 ; c1 , c2 ; −iλz) =

1

0

we have the equation 3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; λz)

1 (2 F1 (a1 , a2 , a3 ; c1 , c2 ; iλz) + 2 F1 (a1 , a2 , a3 ; c1 , c2 ; −iλz)) 2  1 = t a1 −1 (1 − t)c1 −a1 −1 2 Supercos 1 (a2 , a3 ; c2 ; λzt)dt,

=

(2.464)

0

from which we establish the following relationship: 2 Supercos 1 (a2 , a3 ; c1 , c2 ; λz) =

1 (2 F1 (a2 , a3 ; c1 , c2 ; iλz) + 2 F1 (a2 , a3 ; c1 , c2 ; −iλz)) (2.465) 2

where a1 , a2 , a3 , c1 , c2 , λ, z ∈ C and Re (c1 ) > Re (a1 ) > 0.

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2.4.2 The series of the hypergeometric supertrigonometric functions with three numerator parameters and two denominator parameters Using the hypergeometric supertrigonometric functions with three numerator parameters and two denominator parameters, we now take into consideration the series of the hypergeometric supertrigonometric functions with three numerator parameters and two denominator parameters. Definition 2.23. The series of the hypergeometric supertrigonometric functions with three numerator parameters and two denominator parameters can be taken in the form  (z) =

∞ 

φ (κ) (3 F2 (a1 , a2 , a3 ; c1 , c2 ; iκz)),

(2.466)

κ=0

which can be represented as  (z) =γ +

∞     ϕ (κ) 3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; κz) κ=1

+

∞ 

(2.467)

   ψ (κ) 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; κz) ,

κ=1

where ϕ (0) − iψ (0) , 2 ϕ (κ) − iψ (κ) φ (κ) = , 2 γ=

(2.468) (2.469)

and ϕ (κ) + iψ (κ) (2.470) 2 are the coefficients of the supertrigonometric functions via Clausen hypergeometric series with three numerator parameters and two denominator parameters. φ (−κ) =

On account of the supertrigonometric functions via Clausen hypergeometric series with three numerator parameters and two denominator parameters, we may restrict ourselves to the series of the following types. Series 1 If  (−z) =  (z), then there exists the series of the form  (z) = γ +

∞     ϕ (κ) 3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; κz) , κ=1

(2.471)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

105

where γ and ϕ (κ) are the coefficients of the supertrigonometric functions via Clausen hypergeometric series. Series 2 If  (−z) = − (z), then there exists the series of the form  (z) =

∞     ψ (κ) 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; κz) ,

(2.472)

κ=1

where ψ (κ) are the coefficients of the hypergeometric supertrigonometric functions via Clausen hypergeometric series with three numerator parameters and two denominator parameters.

2.4.3 The hypergeometric superhyperbolic functions with three numerator parameters and two denominator parameters In terms of the Clausen hypergeometric series with three numerator parameters and two denominator parameters, we now take into consideration the hypergeometric superhyperbolic functions with three numerator parameters and two denominator parameters. Definition 2.24. The hypergeometric superhyperbolic sine via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; z) =

∞  (a1 )2n+1 (a3 )2n+1 (a3 )2n+1 n=0

(c1 )2n+1 (c2 )2n+1

z2n+1 , (2n + 1)!

(2.473)

where a1 , a2 , a3 , c1 , c2 , z ∈ C and Re (c1 ) > Re (a1 ) > 0. The convergences for the hypergeometric superhyperbolic sine via Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows: The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it conq  p verges absolutely for Re k=1 ck − k=1 ak > 0. From Eq. (2.419) we derive the following equalities: 0 (z) =

(a1 )1 (a3 )1 (a3 )1 z , (c1 )1 (c2 )1 1!

(2.474)

1 (z) =

(a1 )3 (a3 )3 (a3 )3 z3 , 3! (c1 )3 (c2 )3

(2.475)

2 (z) =

(a1 )5 (a3 )5 (a3 )5 z5 , 5! (c1 )5 (c2 )5

(2.476)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

3 (z) =

(a1 )7 (a3 )7 (a3 )7 z7 , (c1 )7 (c2 )7 71!

(2.477)

4 (z) =

(a1 )9 (a3 )9 (a3 )9 z9 , 9! (c1 )9 (c2 )9

(2.478)

(a1 )11 (a3 )11 (a3 )11 z11 , 11! (c1 )11 (c2 )11

(2.479)

and 5 (z) = where |z| < 1. Definition 2.25. The hypergeometric superhyperbolic cosine via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; z) =

∞  (a1 )2n (a3 )2n (a3 )2n z2n , (c1 )2n (c2 )2n (2n)!

(2.480)

n=0

where a1 , a2 , a3 , c1 , c2 , z ∈ C and Re (c1 ) > Re (a1 ) > 0. The convergences for the hypergeometric superhyperbolic cosine via Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows: The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it conq  p verges absolutely for Re c − a > 0. k=1 k k=1 k From Eq. (2.419) we can get the following equalities: 0 (z) = 1,

(2.481)

1 (z) =

(a1 )2 (a3 )2 (a3 )2 z2 , 2! (c1 )2 (c2 )2

(2.482)

2 (z) =

(a1 )4 (a3 )4 (a3 )4 z4 , 4! (c1 )4 (c2 )4

(2.483)

3 (z) =

(a1 )6 (a3 )6 (a3 )6 z6 , 6! (c1 )6 (c2 )6

(2.484)

4 (z) =

(a1 )8 (a3 )8 (a3 )8 z8 , 8! (c1 )8 (c2 )8

(2.485)

(a1 )10 (a3 )10 (a3 )10 z10 , 10! (c1 )10 (c2 )10

(2.486)

and 5 (z) = where |z| < 1.

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

107

Definition 2.26. The hypergeometric superhyperbolic tangent via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; z) , (2.487) 3 Supertanh2 (a1 , a2 , a3 ; c1 , c2 ; z) = 3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; z) where a1 , a2 , a3 , c1 , c2 , z ∈ C and Re (c1 ) > Re (a1 ) > 0. The convergences for the hypergeometric superhyperbolic tangent via Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows: The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it con q p verges absolutely for Re k=1 ck − k=1 ak > 0. Definition 2.27. The hypergeometric superhyperbolic cotangent via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supercoth2 (a1 , a2 , a3 ; c1 , c2 ; z) =

3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; z) 3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; z)

,

(2.488)

where a1 , a2 , a3 , c1 , c2 , z ∈ C and Re (c1 ) > Re (a1 ) > 0. The convergences for the hypergeometric superhyperbolic cotangent via Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows: The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it con q p > 0. c − a verges absolutely for Re k k k=1 k=1 Definition 2.28. The hypergeometric superhyperbolic secant via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supersech2 (a1 , a2 , a3 ; c1 , c2 ; z) =

1 , Supercosh (a 3 2 1 , a2 , a3 ; c1 , c2 ; z)

(2.489)

where a1 , a2 , a3 , c1 , c2 , z ∈ C and Re (c1 ) > Re (a1 ) > 0. The convergences for the hypergeometric superhyperbolic secant via Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows: The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it con q p > 0. c − a verges absolutely for Re k=1 k k=1 k

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Definition 2.29. The hypergeometric superhyperbolic cosecant via Clausen hypergeometric series with three numerator parameters and two denominator parameters is defined as 3 Supercosech2 (a1 , a2 , a3 ; c1 , c2 ; z) =

1 , Supersinh (a 3 2 1 , a2 , a3 ; c1 , c2 ; z)

(2.490)

where a1 , a2 , a3 , c1 , c2 , z ∈ C and Re (c1 ) > Re (a1 ) > 0. The convergences for the hypergeometric superhyperbolic cosecant via Clausen hypergeometric series with three numerator parameters and two denominator parameters are given as follows: The series converges absolutely for |z| < 1, diverges for |z| > 1, and for |z| = 1, it conq  p verges absolutely for Re > 0. c − a k=1 k k=1 k Definition 2.30. The set of the hypergeometric superhyperbolic functions via Clausen hypergeometric series with three numerator parameters and two denominator parameters containing the hypergeometric superhyperbolic sine 3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; z), hypergeometric superhyperbolic cosine 3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; z), hypergeometric superhyperbolic tangent 3 Supertanh2 (a1 , a2 , a3 ; c1 , c2 ; z), hypergeometric superhyperbolic cotangent 3 Supercoth2 (a1 , a2 , a3 ; c1 , c2 ; z), hypergeometric superhyperbolic secant and hypergeometric superhyperbolic cosecant 3 Supersech2 (a1 , a2 , a3 ; c1 , c2 ; z), Supercosech , a , a ; c , c ; z) is called the hypergeometric superhyperbolic functions. (a 3 2 1 2 3 1 2 By means of the hypergeometric superhyperbolic functions via Clausen hypergeometric series with three numerator parameters and two denominator parameters, we derive their properties as follows. Property 2.9. Let a1 , a2 , a3 , c1 , c2 , λ, z ∈ C and Re (c1 ) > Re (a1 ) > 0. Then we have 3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; λz)

1 (3 F2 (a1 , a2 , a3 ; c1 , c2 ; λz) − 3 F2 (a1 , a2 , a3 ; c1 , c2 ; −λz)) 2 ∞  (a1 )2n+1 (a3 )2n+1 (a3 )2n+1 (λz)2n+1 , = (c1 )2n+1 (c2 )2n+1 (2n + 1)! =

(2.491)

n=0

3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; λz)

1 (3 F2 (a1 , a2 , a3 ; c1 , c2 ; λz) + 3 F2 (a1 , a2 , a3 ; c1 , c2 ; −λz)) 2 ∞  (a1 )2n (a3 )2n (a3 )2n (λz)2n , = (c1 )2n (c2 )2n (2n)! =

(2.492)

n=0

3 Supertanh2 (a1 , a2 , a3 ; c1 , c2 ; λz) =

3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; λz) 3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; λz)

,

(2.493)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

3 Supercoth2 (a1 , a2 , a3 ; c1 , c2 ; λz) = 3 Supersech2 (a1 , a2 , a3 ; c1 , c2 ; λz) =

3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; λz)

109

,

(2.494)

1 , Supercosh , (a 3 2 1 a2 , a3 ; c1 , c2 ; λz)

(2.495)

3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; λz)

and 3 Supercosech2 (a1 , a2 , a3 ; c1 , c2 ; λz) =

1 . Supersinh , (a 3 2 1 a2 , a3 ; c1 , c2 ; λz)

(2.496)

Property 2.10. Suppose that a1 , a2 , a3 , c1 , c2 , z ∈ C and Re (c1 ) > Re (a1 ) > 0. Then we have 3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; −z) = −3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; z),

(2.497)

3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; −z) = 3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; z) ,

(2.498)

3 Supertanh2 (a1 , a2 , a3 ; c1 , c2 ; −z) = −3 Supertanh2 (a1 , a2 , a3 ; c1 , c2 ; z),

(2.499)

3 Supercoth2 (a1 , a2 , a3 ; c1 , c2 ; −z) = −3 Supercoth2 (a1 , a2 , a3 ; c1 , c2 ; z),

(2.500)

3 Supersech2 (a1 , a2 , a3 ; c1 , c2 ; −z) = 3 Supersech2 (a1 , a2 , a3 ; c1 , c2 ; z) ,

(2.501)

3 Supercosech2 (a1 , a2 , a3 ; c1 , c2 ; −z) = −3 Supercosech2 (a1 , a2 , a3 ; c1 , c2 ; z).

(2.502)

and

Using the hypergeometric superhyperbolic functions via Clausen hypergeometric series with three numerator parameters and two denominator parameters, we obtain the following theorems. Theorem 2.132. If a1 , a2 , a3 , c1 , c2 , λ, z ∈ C, Re (c1 ) > Re (a1 ) > 0, and |z| < 1, then we have 3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; λz)



=

1

t a1 −1 (1 − t)c1 −a1 −1 2 Supersinh1 (a2 , a3 ; c2 ; λzt)dt,

(2.503)

0

where 2 Supersinh1 (a2 , a3 ; c2 ; λzt) =

1 (2 F1 (a2 , a3 ; c2 ; λzt) − 2 F1 (a2 , a3 ; c2 ; λzt)) . 2

(2.504)

Proof. By using the equalities 

1

3 F2 (a1 , a2 , a3 ; c1 , c2 ; λz) =

t a1 −1 (1 − t)c1 −a1 −1 2 F1 (a2 , a3 ; c2 ; λzt)dt

(2.505)

t a1 −1 (1 − t)c1 −a1 −1 2 F1 (a2 , a3 ; c2 ; −λzt)dt

(2.506)

0

and 

1

3 F2 (a1 , a2 , a3 ; c1 , c2 ; −λz) = 0

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

we reduce to the equality 3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; λz)

1 (2 F1 (a1 , a2 , a3 ; c1 , c2 ; λz) − 2 F1 (a1 , a2 , a3 ; c1 , c2 ; −λz)) 2i  1 = t a1 −1 (1 − t)c1 −a1 −1 2 Supersinh1 (a2 , a3 ; c2 ; λzt)dt,

=

(2.507)

0

in which 2 Supersinh1 (a2 , a3 ; c1 , c2 ; λz) =

1 (2 F1 (a2 , a3 ; c1 , c2 ; λz) − 2 F1 (a2 , a3 ; c1 , c2 ; −λz)) (2.508) 2

with a1 , a2 , a3 , c1 , c2 , λ, z ∈ C and Re (c1 ) > Re (a1 ) > 0. Theorem 2.133. If a1 , a2 , a3 , c1 , c2 , λ, z ∈ C and Re (c1 ) > Re (a1 ) > 0, then we have 

1

3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; λz) =

t a1 −1 (1 − t)c1 −a1 −1 2 Supercosh1 (a2 , a3 ; c2 ; λzt)dt,

0

(2.509)

where 1 (2 F1 (a2 , a3 ; c2 ; λz) + 2 F1 (a2 , a3 ; c2 ; −λz)) . 2

2 Supercosh1 (a2 , a3 ; c2 ; λz) =

(2.510)

Proof. Using the relation 2 Supercosh1 (a2 , a3 ; c1 , c2 ; λz) =

1 (2 F1 (a2 , a3 ; c1 , c2 ; λz) + 2 F1 (a2 , a3 ; c1 , c2 ; −λz)) (2.511) 2

together with the equalities 

1

3 F2 (a1 , a2 , a3 ; c1 , c2 ; λz) =

t a1 −1 (1 − t)c1 −a1 −1 2 F1 (a2 , a3 ; c2 ; λzt)dt

(2.512)

t a1 −1 (1 − t)c1 −a1 −1 2 F1 (a2 , a3 ; c2 ; −λzt)dt,

(2.513)

0

and  3 F2 (a1 , a2 , a3 ; c1 , c2 ; −λz) =

1

0

we easily verify that 3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; λz)

1 (2 F1 (a1 , a2 , a3 ; c1 , c2 ; λz) + 2 F1 (a1 , a2 , a3 ; c1 , c2 ; −λz)) 2  1 = t a1 −1 (1 − t)c1 −a1 −1 2 Supercosh1 (a2 , a3 ; c2 ; λzt)dt =

0

where a1 , a2 , a3 , c1 , c2 , λ, z ∈ C and Re (c1 ) > Re (a1 ) > 0.

(2.514)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

111

Property 2.11. Suppose that a1 , a2 , a3 , c1 , c2 , z ∈ C and Re (c1 ) > Re (a1 ) > 0. Then we have 3 Supersinh2 (a1 , a2 , a3 ; c1 , c2 ; −z) = −i 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; iz) ,

(2.515)

3 Supercosh2 (a1 , a2 , a3 ; c1 , c2 ; −z) = 3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; iz) ,

(2.516)

3 Supertanh2 (a1 , a2 , a3 ; c1 , c2 ; −z) = −i 3 Supertan2 (a1 , a2 , a3 ; c1 , c2 ; iz),

(2.517)

3 Supercoth2 (a1 , a2 , a3 ; c1 , c2 ; −z) = i 3 Supercot 2 (a1 , a2 , a3 ; c1 , c2 ; iz) ,

(2.518)

3 Supersech2 (a1 , a2 , a3 ; c1 , c2 ; −z) = 3 Supersec 2 (a1 , a2 , a3 ; c1 , c2 ; iz) ,

(2.519)

3 Supercosech2 (a1 , a2 , a3 ; c1 , c2 ; −z) = i 3 Supercosec 2 (a1 , a2 , a3 ; c1 , c2 ; iz).

(2.520)

and

Property 2.12. Suppose that an , cn , μ, λ, z ∈ C, n = 1, 2, 3, i = Then the hypergeometric series

√ −1, and Re (c1 ) > Re (a1 ) > 0.

μ3 F2 (a1 , a2 , a2 ; c1 , c2 ; iλz) =μ

∞  (a1 )n (a2 )n (a3 )n (iλz)n n=0

(c1 )n (c2 )n

n!

(2.521)

= μ3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; λz) + iμ3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; λz) is a solution of the differential equation (Q (2, μ, λ, cn ) ϕ)2 (z) − (P (3, μ, λ, an ) ϕ)2 (z) = 0,

(2.522)

where (Q (2, μ, λ, cn ) ϕ)2 (z)  2   d ((Q (2, μ, λ, cn ) ϕ) (z)) 2 d + (cn − 1) ((Q (2, μ, λ, cn ) ϕ) (z)) z =μ z dz dz

(2.523)

(P (3, μ, λ, an ) ϕ)2 (z)  3   d ((P (3, μ, λ, an ) ϕ) (z)) z + an ((P (3, μ, λ, an ) ϕ) (z)) = μ3 λz dz

(2.524)

n=1

and

n=1

with

112

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

(Q (2, μ, λ, cn ) ϕ) (z)  2   dϕ (z) 2 d z + (cn − 1) ϕ (z) =μ z dz dz n=1 2  

 d 2 d =μ z z + (cn − 1) ϕ (z) dz dz

(2.525)

n=1

and (P (3, μ, λ, an ) ϕ) (z)  3   dϕ (z) z = μ3 λz + an ϕ (z) dz n=1

= μ3 λz

(2.526)

3  

  d z + an ϕ (z) . dz

n=1

Proof. Using the equality μ3 F2 (a1 , a2 , a2 ; c1 , c2 ; iλz) = μ3 Supercos 2 (a1 , a2 , a3 ; c1 , c2 ; λz) + μi 3 Supersin2 (a1 , a2 , a3 ; c1 , c2 ; λz) ,

(2.527)

we reduce to (Q (3, μ, λ, cn ) ϕ) (z) − i (P (2, μ, λ, an ) ϕ) (z) = 0.

(2.528)

Substituting Eq. (2.527) into Eq. (2.528), we present the equation of the form (Q (q, μ, λ, cn ) ϕ)2 (z) + (P (p, μ, λ, an ) ϕ)2 (z) = 0, √ where an , cn , μ, λ, z ∈ C, n, p, q ∈ N0 , i = −1, and Re (c1 ) > Re (a1 ) > 0. Consequently, we finish the proof.

(2.529)

2.4.4 Applications to the dilogarithm and related functions Perform similar operations on the dilogarithm, we have the following results. Using the equality [25]  z  1 Li2 (z) = F 1; 2; t) dt = z (1, 2 1 2 F1 (1, 1; 2; zt) dt = z3 F2 (1, 1, 1; 2, 2; z), 0

(2.530)

0

we define the dilogarithm by the series [25] Li2 (z) =

∞ n  z κ=1

where |t| ≤ 1.

n2

 =− 0

z

log (1 − t) dt, t

(2.531)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

113

By the equality 3 F2 (1, 1, 1; 2, 2; z) =

Li2 (z) z

(2.532)

we set up the hypergeometric supertrigonometric and superhyperbolic functions via the dilogarithm as follows: 3 Supersin2 (1, 1, 1; 2, 2; z) ∞  (1)2n+1 (1)2n+1 (1)2n+1

=

(2)2n+1 (2)2n+1

n=0

=−

(−1)n z2n+1 (2n + 1)!

(2.533)

Li2 (iz) + Li2 (−iz) , 2z

3 Supercos 2 (1, 1, 1; 2, 2; z) ∞  (a1 )2n (a3 )2n (a3 )2n (−1)n z2n

=

(c1 )2n (c2 )2n

n=0

=

(2n)!

Li2 (iz) − Li2 (−iz) , 2zi

3 Supertan2 (1, 1, 1; 2, 2; z) = −i

3 Supersec 2 (1, 1, 1; 2, 2; z) =

Li2 (iz) + Li2 (−iz) , Li2 (iz) − Li2 (−iz)

(2.536)

2zi , Li2 (iz) − Li2 (−iz)

(2.537)

3 Supercosec 2 (1, 1, 1; 2, 2; z) = −

2z , Li2 (iz) + Li2 (−iz)

3 Supersinh2 (1, 1, 1; 2, 2; z) ∞  (1)2n+1 (1)2n+1 (1)2n+1

=

n=0

(2.535)

Li2 (iz) − Li2 (−iz) , Li2 (iz) + Li2 (−iz)

3 Supercot 2 (1, 1, 1; 2, 2; z) = i

=

(2.534)

(2)2n+1 (2)2n+1

z2n+1 (2n + 1)!

(2.538)

(2.539)

1 (Li2 (z) − Li2 (−z)) , 2z 3 Supercosh2 (1, 1, 1; 2, 2; z) ∞  (1)2n (1)2n (1)2n z2n

=

n=0

(2)2n (2)2n

(2n)!

1 (Li2 (z) + Li2 (−z)) , 2z Li2 (z) − Li2 (−z) , 3 Supertanh2 (1, 1, 1; 2, 2; z) = Li2 (z) + Li2 (−z)

(2.540)

=

(2.541)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

3 Supercoth2 (1, 1, 1; 2, 2; z) =

Li2 (z) + Li2 (−z) , Li2 (z) − Li2 (−z)

(2.542)

3 Supersech2 (1, 1, 1; 2, 2; z) =

2z , Li2 (z) + Li2 (−z)

(2.543)

and 3 Supercosech2 (1, 1, 1; 2, 2; z) =

2z , Li2 (z) − Li2 (−z)

(2.544)

where z ∈ C, n ∈ N0 , and |z| < 1. On account of the hypergeometric supertrigonometric and superhyperbolic functions via the dilogarithm, we obtain the following theorems. Theorem 2.134. If z ∈ C\ {0} and |z| < 1, then we have 

1

3 Supersin2 (1, 1, 1; 2, 2; z) =

2 Supersin1 (1, 1; 2; zt) dt,

(2.545)

0

where 2 Supersin1 (1, 1; 2; λz) =

1 (2 F1 (1, 1; 2; iz) − 2 F1 (1, 1; 2; iz)) . 2i

(2.546)

Theorem 2.135. If z ∈ C\ {0} and |z| < 1, then we have 

1

3 Supercos 2 (1, 1, 1; 2, 2; z) =

2 Supercos 1 (1, 1; 2; zt) dt,

(2.547)

0

where 2 Supercos 1 (1, 1; 2; z) =

1 (2 F1 (1, 1; 2; iz) + 2 F1 (1, 1; 2; −iz)) . 2

(2.548)

Theorem 2.136. If z ∈ C\ {0} and |z| < 1, then we have 

1

3 Supersinh2 (1, 1, 1; 2, 2; z) =

2 Supersinh1 (1, 1; 2; zt)dt,

(2.549)

0

where 2 Supersinh1 (1, 1; 2; z) =

1 (2 F1 (1, 1; 2; z) − 2 F1 (1, 1; 2; −z)) . 2

(2.550)

Theorem 2.137. If z ∈ C\ {0} and |z| < 1, then we have 

1

3 Supercosh2 (1, 1, 1; 2, 2; z) =

2 Supercosh1 (1, 1; 2; zt)dt,

(2.551)

0

where 2 Supercosh1 (1, 1; 2; z) =

1 (2 F1 (1, 1; 2; z) + 2 F1 (1, 1; 2; −z)) . 2

(2.552)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

115

2.5 Analytic number theory via Clausen hypergeometric functions In this section, we present the theorems for the number formulae via Clausen hypergeometric functions as follows.

2.5.1 Special formulae via Clausen hypergeometric functions Without proof, we now consider the following theorems.

  Theorem 2.138. If Re > 0, Re > 0, Re > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , (α) (β) (a ) 1   Re cq > 0, and p, q ∈ N0 , then 

  t α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; t dt 0   a1 , · · · , ap , α; 1 . = p+1 Fq+1 c1 , · · · , cq , α + β;

1 B (α, β)

1

Theorem 2.139. If Re (α) > 0 and Re (β) > 0, then    1 1 α; α−1 β−1 t 1 . t e dt = 1 F1 (1 − t) α + β; B (α, β) 0 Theorem 2.140. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then    1 1 a1 , α; α−1 β−1 1 . t (1 − t) 1 F1 (a1 ; c1 ; t)dt = 2 F2 c1 , α + β; B (α, β) 0

(2.553)

(2.554)

(2.555)

Theorem 2.141. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and Re (c2 ) > 0, then    1 1 a1 , a2 , α; 1 . (2.556) t α−1 (1 − t)β−1 2 F2 (a1 , a2 ; c1 , c2 ; t)dt = 3 F3 c1 , c2 , α + β; B (α, β) 0 Theorem 2.142. If Re (α) > 0, Re (β) > 0, and Re (c) > 0, then    1 1 α; 1 . t α−1 (1 − t)β−1 0 F1 (−; c; t)dt = 1 F2 c, α + β; B (α, β) 0

(2.557)

Theorem 2.143. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, and Re (c2 ) > 0, then    1 1 a1 , α; α−1 β−1 1 . (2.558) t (1 − t) 1 F2 (a1 ; c1 , c2 ; t)dt = 2 F3 c1 , c2 , α + β; B (α, β) 0 Theorem 2.144. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then    1 1 a1 , α; α−1 β−1 1 . t (1 − t) 1 F0 (a1 ; −; t)dt = 2 F1 α + β; B (α, β) 0

(2.559)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 2.145. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then 1 B (α, β)





1

t

α−1

(1 − t)

β−1

1 F0 (a1 ; −; 2t)dt

= 2 F1

0

 a1 , α; 2 . α + β;

(2.560)

Theorem 2.146. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then 1 B (α, β)



1

t α−1 (1 − t)β−1 (1 − 3t)−a dt = 2 F1



0

 a1 , α; 3 . α + β;

(2.561)

  Theorem 2.147. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, and p, q ∈ N0 , then we have 



t α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; t 2 dt 0

a1 , · · · , ap , α2 , α+1 2 ; = p+2 Fq+2 α+β+1 1 . c1 , · · · , cq , α+β ; 2 , 2

1 B (α, β)

1

(2.562)

Theorem 2.148. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then 1 B (α, β)



1

t

α−1

(1 − t)

β−1



2 a dt = 3 F3 F ; c ; t 1 1 1 1





a1 , α2 , α+1 2 ;

1 .

α+β+1 c1 , α+β ; 2 , 2

0

(2.563)

Theorem 2.149. If Re (α) > 0 and Re (β) > 0, then 1 B (α, β)





1

t

α−1

(1 − t)

β−1 t 2

e dt = 2 F2

0

α α+1 2, 2 ; α+β α+β+1 1 ; 2 , 2

(2.564)

.

Theorem 2.150. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then 

1 B (α, β)

1

t

α−1

(1 − t)

β−1

1 F0

a1 ; −; t

2

dt = 3 F2

0

a1 , α2 , α+1 2 ;



α+β α+β+1 1 ; 2 , 2

(2.565)

.

Theorem 2.151. If Re (α) > 0, Re (β) > 0, and Re (c1 ) > 0, then 1 B (α, β)



1

t

α−1

(1 − t)

β−1

0 F1

−; c1 ; t dt = 2 F3 2

0





α α+1 2, 2 ; α+β α+β+1 1 c1 , 2 , 2 ;

.

(2.566)

.

(2.567)

Theorem 2.152. If Re (α) > 0 and Re (β) > 0, then 1 B (α, β)





1

t 0

α−1

β−1 t 3

(1 − t)

e dt = 3 F3

α α+1 α+2 3, 3 , 3 ; 1 α+β α+β+1 α+β+2 , 3 ; 3 , 3



Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

117

Theorem 2.153. If Re (α) > 0 and Re (β) > 0, then 1 B (α, β)





1

t

α−1

(1 − t)

β−1 t κ

0

e dt = κ Fκ

α α+1 α+κ−1 ; κ , κ ,··· , κ α+β α+β+1 α+β+κ−1 1 ,··· , ; κ , κ κ

,

(2.568)

where κ ∈ N.   Theorem 2.154. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, and p, q ∈ N0 , then we have   t dt t (2 − t) p Fq a 1 , · · · , ap ; c1 , · · · , cq ; 2 0    1 1 1+t α−1 β−1 = dt (1 + t) (1 − t) p Fq a 1 , · · · , ap ; c1 , · · · , cq ; B (α, β) −1 2   a1 , · · · , ap , α; 1 . = 2α+β−1 × p+1 Fq+1 c1 , · · · , cq , α + β;

1 B (α, β)



2

α−1

β−1

(2.569)

  Theorem 2.155. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, and p, q ∈ N0 , then we have  2   1 t α−1 (2 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; t dt B (α, β) 0  1   1 = (1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; 1 + t dt B (α, β) −1   a1 , · · · , ap , α; 2 . = 2α+β−1 × p+1 Fq+1 c1 , · · · , cq , α + β;

(2.570)

  Theorem 2.156. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, and p, q ∈ N0 , then we have  2   1 t α−1 (2 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; 2t dt B (α, β) 0  1   1 = (1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; 2 (1 + t) dt B (α, β) −1   a1 , · · · , ap , α; α+β−1 4 . × p+1 Fq+1 =2 c1 , · · · , cq , α + β;

(2.571)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 2.157. If Re (α) > 0 and Re (β) > 0, then  2 t 1 t α−1 (2 − t)β−1 e 2 dt B (α, β) 0 √  1 t e = (1 + t)α−1 (1 − t)β−1 e 2 dt B (α, β) −1   α; α+β−1 =2 × 1 F1 1 . α + β; Theorem 2.158. If Re (α) > 0 and Re (β) > 0, then  2 1 t α−1 (2 − t)β−1 et dt B (α, β) 0  1 e = (1 + t)α−1 (1 − t)β−1 et dt B (α, β) −1   α; α+β−1 × 1 F1 =2 2 . α + β; Theorem 2.159. If Re (α) > 0 and Re (β) > 0, then  2 1 t α−1 (2 − t)β−1 e2t dt B (α, β) 0  1 e2 = (1 + t)α−1 (1 − t)β−1 e2t dt B (α, β) −1   α; = 2α+β−1 × 1 F1 4 . α + β; Theorem 2.160. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then    2 1 t α−1 β−1 dt t (2 − t) 1 F1 a 1 ; c1 ; B (α, β) 0 2    1 1 t +1 α−1 β−1 = dt (1 + t) (1 − t) 1 F1 a 1 ; c1 ; B (α, β) −1 2   a1 , α; = 2α+β−1 × 2 F2 1 . c1 , α + β; Theorem 2.161. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then  2 1 t α−1 (2 − t)β−1 1 F1 (a1 ; c1 ; t)dt B (α, β) 0  1 1 = (1 + t)α−1 (1 − t)β−1 1 F1 (a1 ; c1 ; t + 1)dt B (α, β) −1   a1 , α; α+β−1 2 . × 2 F2 =2 c1 , α + β;

(2.572)

(2.573)

(2.574)

(2.575)

(2.576)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

119

Theorem 2.162. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then   3t dt t (2 − t) 1 F1 a 1 ; c1 ; 2 0    1 1 3 = (1 + t)α−1 (1 − t)β−1 1 F1 a1 ; c1 ; (t + 1) dt B (α, β) −1 2   a1 , α; 3 . = 2α+β−1 × 2 F2 c1 , α + β;

1 B (α, β)



2

α−1

β−1

(2.577)

Theorem 2.163. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then  2 1 t α−1 (2 − t)β−1 1 F1 (a1 ; c1 ; 2t)dt B (α, β) 0  1 1 = (t + 1)α−1 (1 − t)β−1 1 F1 (a1 ; c1 ; 2 (1 + t))dt B (α, β) −1   a1 , α; α+β−1 4 . × 2 F2 =2 c1 , α + β;

(2.578)

Theorem 2.164. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and Re (c2 ) > 0, then   t dt t (2 − t) 2 F2 a 1 , a2 ; c1 , c2 ; 2 0    1 1 1 = (1 + t)α−1 (1 − t)β−1 2 F2 a1 , a2 ; c1 , c2 ; (1 + t) dt B (α, β) −1 2   a1 , a2 , α; = 2α+β−1 × 3 F3 1 . c1 , c2 , α + β;

1 B (α, β)



2

α−1

β−1

(2.579)

Theorem 2.165. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and Re (c2 ) > 0, then  2 1 t α−1 (2 − t)β−1 2 F2 (a1 , a2 ; c1 , c2 ; t)dt B (α, β) 0  1 1 = (1 + t)α−1 (1 − t)β−1 2 F2 (a1 , a2 ; c1 , c2 ; 1 + t)dt B (α, β) −1   a1 , a2 , α; 2 . = 2α+β−1 × 3 F3 c1 , c2 , α + β;

(2.580)

Theorem 2.166. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and Re (c2 ) > 0, then

120

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions  2 1 t α−1 (2 − t)β−1 2 F2 (a1 , a2 ; c1 , c2 ; 2t)dt B (α, β) 0  1 1 = (1 + t)α−1 (1 − t)β−1 2 F2 (a1 , a2 ; c1 , c2 ; 2 (1 + t))dt B (α, β) −1   a1 , a2 , α; 4 . = 2α+β−1 × 3 F3 c1 , c2 , α + β;

(2.581)

Theorem 2.167. If Re (α) > 0, Re (β) > 0, and Re (c) > 0, then   t dt t (2 − t) 0 F1 −; c; 2 0    1 1 1 = (1 + t)α−1 (1 − t)β−1 0 F1 −; c; (1 + t) dt B (α, β) −1 2   α; 1 . = 2α+β−1 × 1 F2 c, α + β;

1 B (α, β)



2

α−1

β−1

(2.582)

Theorem 2.168. If Re (α) > 0, Re (β) > 0, and Re (c) > 0, then  2 1 t α−1 (2 − t)β−1 0 F1 (−; c; t)dt B (α, β) 0  1 1 = (1 + t)α−1 (1 − t)β−1 0 F1 (−; c; 1 + t)dt B (α, β) −1   α; α+β−1 × 1 F2 =2 2 . c, α + β;

(2.583)

Theorem 2.169. If Re (α) > 0, Re (β) > 0, and Re (c) > 0, then  2 1 t α−1 (2 − t)β−1 0 F1 (−; c; 2t)dt B (α, β) 0  1 1 = (1 + t)α−1 (1 − t)β−1 0 F1 (−; c; 2 (1 + t))dt B (α, β) −1   α; α+β−1 × 1 F2 =z 4 . c, α + β;

(2.584)

Theorem 2.170. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, and Re (c2 ) > 0, then   t dt t α−1 (2 − t)β−1 1 F2 a1 ; c1 , c2 ; 2 0    1 1 1 α−1 β−1 = (1 + t) (1 − t) 1 F2 a1 ; c1 , c2 ; (1 + t) dt B (α, β) −1 2   a1 , α; 1 . = 2α+β−1 × 2 F3 c1 , c2 , α + β;

1 B (α, β)



2

(2.585)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

121

Theorem 2.171. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, and Re (c2 ) > 0, then  2 1 t α−1 (2 − t)β−1 1 F2 (a1 ; c1 , c2 ; t)dt B (α, β) 0  1 1 (2.586) = (1 + t)α−1 (1 − t)β−1 1 F2 (a1 ; c1 , c2 ; 1 + t)dt B (α, β) −1   a1 , α; α+β−1 4 . =2 × 2 F3 c1 , c2 , α + β; Theorem 2.172. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, and Re (c2 ) > 0, then  2 1 t α−1 (2 − t)β−1 1 F2 (a1 ; c1 , c2 ; 2t)dt B (α, β) 0  1 1 (2.587) = (1 + t)α−1 (1 − t)β−1 1 F2 (a1 ; c1 , c2 ; 2 (1 + t))dt B (α, β) −1   a1 , α; α+β−1 4 . × 2 F3 =2 c1 , c2 , α + β; Theorem 2.173. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then    2 1 t α−1 β−1 dt t (2 − t) 1 F0 a1 ; −; B (α, β) 0 2    1 1 1 = (1 + t)α−1 (1 − t)β−1 1 F0 a1 ; −; (1 + t) dt B (α, β) −1 2   a1 , α; = 2α+β−1 × 2 F1 1 . α + β; Theorem 2.174. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then  2 1 t α−1 (2 − t)β−1 1 F0 (a1 ; −; t)dt B (α, β) 0  1 1 = (1 + t)α−1 (1 − t)β−1 1 F0 (a1 ; −; 1 + t)dt B (α, β) −1   a1 , α; = 2α+β−1 × 2 F1 1 . α + β; Theorem 2.175. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then  2 1 t α−1 (2 − t)β−1 1 F0 (a1 ; −; 2t)dt B (α, β) 0  1 1 = (1 + t)α−1 (1 − t)β−1 1 F0 (a1 ; −; 2 (1 + t))dt B (α, β) −1   a1 , α; α+β−1 4 . × 2 F1 =2 α + β;

(2.588)

(2.589)

(2.590)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

  Theorem 2.176. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, and p, q ∈ N0 , then we have   t2 dt t (2 − t) p Fq a 1 , · · · , ap ; c1 , · · · , cq ; 4 0    1 1 1 = (1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; (1 + t)2 dt B (α, β) −1 4

α α+1 a1 , · · · , ap , 2 , 2 ; = 2α+β−1 × p+2 Fq+2 α+β+1 1 . ; c1 , · · · , cq , α+β 2 , 2

1 B (α, β)



2

α−1

β−1

(2.591)

  Theorem 2.177. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, and p, q ∈ N0 , then we have  2

1 t α−1 (2 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; t 2 dt B (α, β) 0  1

1 = (1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; (1 + t)2 dt B (α, β) −1

α α+1 , · · · , a , , ; a 1 p 2 2 = 2α+β−1 × p+2 Fq+2 4 . α+β+1 c1 , · · · , cq , α+β ; 2 , 2

(2.592)

  Theorem 2.178. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, and p, q ∈ N0 , then we have  2

1 t α−1 (2 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; 2t 2 dt B (α, β) 0  1

1 = (1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; 2 (1 + t)2 dt B (α, β) −1

a1 , · · · , ap , α2 , α+1 2 ; α+β−1 =2 × p+2 Fq+2 α+β+1 8 . c1 , · · · , cq , α+β ; 2 , 2

(2.593)

Theorem 2.179. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then we have 1 B (α, β)

  t2 dt t α−1 (2 − t)β−1 1 F1 a1 ; c1 ; 4 0 



1 = B (α, β)

2

1

−1

(1 + t)

= 2α+β−1 × 3 F3



α−1

(1 − t)

β−1

a1 , α2 , α+1 2 ; α+β+1 ; c1 , α+β 2 , 2

1 F1

1 .

(1 + t)2 a1 ; c1 ; dt 4

(2.594)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

123

Theorem 2.180. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then we have  2

1 t α−1 (2 − t)β−1 1 F1 a1 ; c1 ; t 2 dt B (α, β) 0  1

1 = (1 + t)α−1 (1 − t)β−1 1 F1 a1 ; c1 ; (1 + t)2 dt B (α, β) −1

α α+1 a , , ; 1 2 2 = 2α+β−1 × 3 F3 α+β+1 4 . c1 , α+β ; 2 , 2

(2.595)

Theorem 2.181. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then we have  2

1 t α−1 (2 − t)β−1 1 F1 a1 ; c1 ; 2t 2 dt B (α, β) 0  1

1 = (1 + t)α−1 (1 − t)β−1 1 F1 a1 ; c1 ; 2 (1 + t)2 dt B (α, β) −1

a1 , α2 , α+1 2 ; α+β−1 =2 × 3 F3 α+β+1 8 . ; c1 , α+β 2 , 2

(2.596)

Theorem 2.182. If Re (α) > 0 and Re (β) > 0, then  2 t2 1 t α−1 (2 − t)β−1 e 4 dt B (α, β) 0  1 (1+t)2 1 = (1 + t)α−1 (1 − t)β−1 e 4 dt B (α, β) −1

α α+1 2, 2 ; α+β−1 =2 × 2 F2 α+β α+β+1 1 . ; 2 , 2

(2.597)

Theorem 2.183. If Re (α) > 0 and Re (β) > 0, then  2 1 2 t α−1 (2 − t)β−1 et dt B (α, β) 0  1 1 2 = (1 + t)α−1 (1 − t)β−1 e(1+t) dt B (α, β) −1

α α+1 2, 2 ; α+β−1 =2 × 2 F2 α+β α+β+1 4 . ; 2 , 2 Theorem 2.184. If Re (α) > 0 and Re (β) > 0, then 1 B (α, β)

 0

2

2

t α−1 (2 − t)β−1 e2t dt

(2.598)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

1 B (α, β)

=



1

2

(1 + t)α−1 (1 − t)β−1 e2(1+t) dt −1

α α+1 2, 2 ; α+β−1 =2 × 2 F2 α+β α+β+1 8 . ; 2 , 2

(2.599)

Theorem 2.185. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then we have 1 B (α, β)

  t2 dt t α−1 (2 − t)β−1 1 F0 a1 ; −; 4 0 



2

1 = B (α, β)

1

−1

(1 + t)

= 2α+β−1 × 3 F2



α−1

(1 − t)

a1 , α2 , α+1 2 ;

β−1



α+β α+β+1 1 ; 2 , 2

1 F0

(1 + t)2 a1 ; −; dt 4

(2.600)

.

Theorem 2.186. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then we have  2

1 t α−1 (2 − t)β−1 1 F0 a1 ; −; t 2 dt B (α, β) 0  1

1 = (1 + t)α−1 (1 − t)β−1 1 F0 a1 ; −; (1 + t)2 dt B (α, β) −1

a1 , α2 , α+1 2 ; α+β−1 =2 × 3 F2 α+β α+β+1 4 . ; 2 , 2

(2.601)

Theorem 2.187. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then we have  2

1 t α−1 (2 − t)β−1 1 F0 a1 ; −; 2t 2 dt B (α, β) 0  1

1 = (1 + t)α−1 (1 − t)β−1 1 F0 a1 ; −; 2 (1 + t)2 dt B (α, β) −1

a1 , α2 , α+1 2 ; α+β−1 =2 × 3 F2 α+β α+β+1 8 . ; 2 , 2

(2.602)

Theorem 2.188. If Re (α) > 0, Re (β) > 0, and Re (c1 ) > 0, then we have   t2 dt t α−1 (2 − t)β−1 0 F1 −; c1 ; 4 0    1 1 1 α−1 β−1 2 = (1 + t) (1 − t) 0 F1 −; c1 ; (1 + t) dt B (α, β) −1 4

α α+1 2, 2 ; = 2α+β−1 × 2 F3 1 . α+β α+β+1 c1 , 2 , 2 ;

1 B (α, β)



2

(2.603)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

Theorem 2.189. If Re (α) > 0, Re (β) > 0, and Re (c1 ) > 0, then we have  2

1 t α−1 (2 − t)β−1 0 F1 −; c1 ; t 2 dt B (α, β) 0  1

1 = (1 + t)α−1 (1 − t)β−1 0 F1 −; c1 ; (1 + t)2 dt B (α, β) −1

α α+1 2, 2 ; α+β−1 =2 × 2 F3 α+β+1 4 . ; c1 , α+β 2 , 2

125

(2.604)

Theorem 2.190. If Re (α) > 0, Re (β) > 0, and Re (c1 ) > 0, then we have  2

1 t α−1 (2 − t)β−1 0 F1 −; c1 ; 2t 2 dt B (α, β) 0  1

1 = (1 + t)α−1 (1 − t)β−1 0 F1 −; c1 ; 2 (1 + t)2 dt (2.605) B (α, β) −1

α α+1 2, 2 ; = 2α+β−1 × 2 F3 8 . α+β α+β+1 c1 , 2 , 2 ;   Theorem 2.191. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , and κ ∈ N, then we have  2 1 −κ κ t α−1 (2 − t)β−1 e2 t dt B (α, β) 0  1 1 κ −κ = (1 + t)α−1 (1 − t)β−1 e2 (1+t) dt (2.606) B (α, β) −1

α α+1 , κ , · · · , α+κ−1 ; κ = 2α+β−1 × κ Fκ α+β κ α+β+1 α+β+κ−1 1 . ,··· , ; κ , κ κ   Theorem 2.192. If Re > 0, Re > 0, Re > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , (α) (β) (a ) 1   Re cq > 0, p, q ∈ N0 , and κ ∈ N, then we have  2 1 κ t α−1 (2 − t)β−1 eλt dt B (α, β) 0  1 1 κ = (1 + t)α−1 (1 − t)β−1 eλ(1+t) dt B (α, β) −1

α α+1 α+κ−1 , , · · · , ; κ κ κ+1 = 2α+β−1 × κ Fκ α+β κ α+β+1 . α+β+κ−1 λ2 , , · · · , ; κ κ κ Theorem 2.193. If Re (α) > 0, Re (β) > 0, and κ, s ∈ N, then we have  1 s κ t α−1 (1 − t)β−1 eλt (1−t) dt 0

(2.607)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

= B (α, β) × κ+s Fκ+s

α

κ,

α+1 α+κ−1 β β+1 , s , s , · · · , β+s−1 ; κ ,··· , κ s α+β α+β+1 α+β+κ+s−1 ; κ+s , κ+s , · · · , κ+s

κ κ ss λ (κ + s)κ+s

(2.608)

,

where λ is a constant. Theorem 2.194. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, and κ, s ∈ N, then we have  1   t α−1 (1 − t)β−1 1 F1 a1 ; c1 ; λt κ (1 − t)s dt 0

= B (α, β)



× 1+κ+s F1+κ+s

α+κ−1 β β+1 , s , s , · · · , β+s−1 ; a1 , ακ , α+1 κ ,··· , κ s α+β α+β+1 α+β+κ+s−1 c1 , κ+s , κ+s , · · · , ; κ+s

(2.609)



κ κ ss λ

,

(κ + s)κ+s

where λ is a constant. Theorem 2.195. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, and κ, s ∈ N, then we have  1   t α−1 (1 − t)β−1 2 F2 a1 , a2 ; c1 , c2 ; λt κ (1 − t)s dt 0

= B (α, β) × 2+κ+s F2+κ+s

α+κ−1 β β+1 , s , s , · · · , β+s−1 ; a1 , a2 , ακ , α+1 κ ,··· , κ s α+β+1 α+β+κ+s−1 ; c1 , c2 , α+β κ+s , κ+s , · · · , κ+s

(2.610)



κ κ ss λ (κ + s)κ+s

,

where λ is a constant. Theorem 2.196. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and κ, s ∈ N, then we have  0

1

t α−1 (z − t)β−1  a dt 1 − λt κ (z − t)s 1

= B (α, β)



× 1+κ+s Fκ+s

α+κ−1 β β+1 , s , s , · · · , β+s−1 ; a1 , ακ , α+1 κ ,··· , κ s α+β α+β+1 α+β+κ+s−1 ; κ+s , κ+s , · · · , κ+s

κ κ ss λ (κ + s)κ+s

(2.611)

,

where λ is a constant. Theorem 2.197. If Re (α) > 0, Re (β) > 0, Re (c1 ) > 0, and κ, s ∈ N, then we have  0

1

  t α−1 (1 − t)β−1 0 F1 −; c1 ; λt κ (1 − t)s dt

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

= B (α, β) × κ+s F1+κ+s

α

κ,

α+1 α+κ−1 β β+1 , s , s , · · · , β+s−1 ; κ ,··· , κ s α+β α+β+1 α+β+κ+s−1 c1 , κ+s , κ+s , · · · , ; κ+s

κ κ ss λ (κ + s)κ+s

127

(2.612)

,

where λ is a constant. Theorem 2.198. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and κ, s ∈ N, then we have  1   t α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; λt κ (1 − t)s dt 0

= B (α, β) × 2+κ+s F1+κ+s

α+κ−1 β β+1 , s , s , · · · , β+s−1 ; a1 , a2 , ακ , α+1 κ ,··· , κ s α+β+1 α+β+κ+s−1 c1 , α+β ; κ+s , κ+s , · · · , κ+s

κ κ ss λ

(2.613)



(κ + s)κ+s

,

where λ is a constant. Theorem 2.199. If Re (α) > 0 and Re (β) > 0, then we have  1

t α−1 (1 − t)β−1 eλt(1−t) dt = B (α, β) × 2 F2

0

λ α+β α+β+1 4 , ; 2 , 2 α, β;

(2.614)

where λ is a constant. Theorem 2.200. If Re (α) > 0 and Re (β) > 0, then we have

 1 α, β2 , β+1 4 2 ; α−1 β−1 λt(1−t)2 t e dt = B (α, β) × 3 F3 α+β α+β+1 α+β+2 λ , (1 − t) , 3 ;9 0 3 , 3

(2.615)

where λ is a constant. Theorem 2.201. If Re (α) > 0 and Re (β) > 0, then we have

 1 β α, α+1 4 2 , 2; α−1 β−1 λ(1−t)t 2 t e dt = B (α, β) × 3 F3 α+β α+β+1 α+β+2 λ , (1 − t) , 3 ;9 0 3 , 3

(2.616)

where λ is a constant. Theorem 2.202. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then we have

 1 α−1 a1 , α, α+1 4 t (1 − t)β−1 2 , β;  a dt = B (α, β) × 4 F3 α+β α+β+1 α+β+2 λ , , 3 ;9 1 − λt (1 − t)2 1 0 3 , 3 where λ is a constant.

(2.617)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 2.203. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then we have

 1 α−1 a1 , α, β, β+1 ; t 4 (1 − t)β−1 2 λ ,  a dt = B (α, β) × 4 F3 α+β α+β+1 α+β+2 1 − λt 2 (1 − t) 1 , , ;9 0 3

3

(2.618)

3

where λ is a constant. Theorem 2.204. If Re (α) > 0, Re (β) > 0, and κ ∈ N, then we have 

1 −1

(1 + t)α−1 (1 − t)β−1 eλ(1+t)

κ (1−t)s

 dt =

2

t α−1 (2 − t)β−1 eλt

κ (2−t)s

dt,

(2.619)

0

where λ is a constant. Theorem 2.205. If Re (α) > 0, Re (β) > 0, and κ, s ∈ N, then we have 

1 −1

(1 + t)α−1 (1 − t)β−1 eλ(1+t)

κ (1−t)s

dt

= B (α, β) 2α+β−1 α α+1

β+s−1 α+κ−1 β β+1 κ s s 2κ+s , , · · · , , , , · · · , ; λκ κ s s s × κ+s Fκ+s κ κα+β α+β+1 , α+β+κ+s−1 (κ + s)κ+s , , · · · , ; κ+s κ+s κ+s

(2.620)

where λ is a constant. Theorem 2.206. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, and κ, s ∈ N, then we have 

1

−1

  (1 + t)α−1 (1 − t)β−1 1 F1 a1 ; c1 ; λ (1 + t)κ (1 − t)s dt

= B (α, β) 2α+β−1

β+s−1 α+κ−1 β β+1 κ s s 2κ+s , · · · , , , , · · · , ; a1 , ακ , α+1 λκ κ κ s s s × 1+κ+s F1+κ+s , α+β+1 α+β+κ+s−1 (κ + s)κ+s c1 , α+β , , · · · , ; κ+s κ+s κ+s

(2.621)

where λ is a constant. Theorem 2.207. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, and κ, s ∈ N, then we have  1   (1 + t)α−1 (1 − t)β−1 2 F2 a1 , a2 ; c1 , c2 ; λ (1 + t)κ (1 − t)s dt −1

= B (α, β) 2α+β−1

α+κ−1 β β+1 , s , s , · · · , β+s−1 ; λκ κ s s 2κ+s a1 , a2 , ακ , α+1 κ ,··· , κ s × 2+κ+s F2+κ+s , α+β+1 α+β+κ+s−1 (κ + s)κ+s ; c1 , c2 , α+β κ+s , κ+s , · · · , κ+s where λ is a constant.

(2.622)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

129

Theorem 2.208. If Re (α) > 0, Re (β) > 0, and κ, s ∈ N, then we have  1 (1 + t)α−1 (1 − t)β−1   dt κ s a1 −1 1 − λ (1 + t) (1 − t) = B (α, β) 2α+β−1

α+κ−1 β β+1 , s , s , · · · , β+s−1 ; λκ κ s s 2κ+s a1 , ακ , α+1 κ ,··· , κ s × 1+κ+s Fκ+s , α+β α+β+1 α+β+κ+s−1 (κ + s)κ+s ; κ+s , κ+s , · · · , κ+s

(2.623)

where λ is a constant. Theorem 2.209. If Re (α) > 0, Re (β) > 0, Re (c1 ) > 0, and κ, s ∈ N, then we have  1   (1 + t)α−1 (1 − t)β−1 0 F1 −; c1 ; λ (1 + t)κ (1 − t)s dt −1

= B (α, β) 2α+β−1 α α+1

α+κ−1 β β+1 , s , s , · · · , β+s−1 ; λκ κ s s 2κ+s κ , κ ,··· , κ s × κ+s F1+κ+s , α+β+1 α+β+κ+s−1 (κ + s)κ+s ; c1 , α+β κ+s , κ+s , · · · , κ+s

(2.624)

where λ is a constant. Theorem 2.210. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and κ, s ∈ N, then we have  1   (1 + t)α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; λ (1 + t)κ (1 − t)s dt −1

= B (α, β) 2α+β−1

β+s−1 α+κ−1 β β+1 κ s s 2κ+s , · · · , , , , · · · , ; a1 , a2 , ακ , α+1 λκ κ κ s s s × 2+κ+s F1+κ+s , α+β+1 α+β+κ+s−1 (κ + s)κ+s c1 , α+β , , · · · , ; κ+s κ+s κ+s

(2.625)

where λ is a constant. Theorem 2.211. If Re (α) > 0 and Re (β) > 0, then we have

 1 α, β; λ α−1 β−1 λ(1+t)(1−t) α+β−1 e dt = B (α, β) 2 , (1 + t) (1 − t) 2 F2 α+β α+β+1 ;4 −1 2 , 2

(2.626)

where λ is a constant. Theorem 2.212. If Re (α) > 0 and Re (β) > 0, then we have  1 2 (1 + t)α−1 (1 − t)β−1 eλ(1+t)(1−t) dt −1

α, β2 , β+1 16 2 ; α+β−1 = B (α, β) 2 × 3 F3 α+β α+β+1 α+β+2 λ , , 3 ; 9 3 , 3 where λ is a constant.

(2.627)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 2.213. If Re (α) > 0 and Re (β) > 0, then we have  1 2 (1 + t)α−1 (1 − t)β−1 eλ(1+t) (1−t) dt −1

β α, α+1 16 2 , 2; α+β−1 λ , = B (α, β) × 2 3 F3 α+β α+β+1 α+β+2 , 3 ; 9 3 , 3

(2.628)

where λ is a constant. Theorem 2.214. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then we have  1 (1 + t)α−1 (1 − t)β−1   dt 2 a1 −1 1 − λ (1 + t) (1 − t)

a1 , α, α+1 16 2 , β; α+β−1 = B (α, β) × 2 λ , 4 F3 α+β α+β+1 α+β+2 , 3 ; 9 3 , 3

(2.629)

where λ is a constant. Theorem 2.215. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then we have  1 (1 + t)α−1 (1 − t)β−1  a1 dt 2 −1 1 − λ (1 + t) (1 − t)

a1 , α, β, β+1 16 2 ; α+β−1 λ , × 4 F3 α+β α+β+1 α+β+2 = B (α, β) 2 , , ; 9 3

3

(2.630)

3

where λ is a constant. Theorem 2.216. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then we have  1 (1 + t)α−1 (1 − t)β−1 a1 dt −1 (1 − λ (1 + t) (1 − t))

, α, β; a 1 = B (α, β) 2α+β−1 × 3 F2 α+β α+β+1 λ , ; 2 , 2

(2.631)

where λ is a constant.

  Theorem 2.217. If Re > 0, Re > 0, Re > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · , (α) (β) (a ) 1   Re cq > 0, p, q ∈ N0 , and s ∈ N, then we have  1   t α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt 0

= B (α, β) × p+s Fq+s where λ is a constant.



β+s−1 a1 , · · · , ap , βs , β+1 ; s ,··· , s λ α+β α+β+1 α+β+s−1 ; c1 , · · · , cq , s , s , · · · , s

(2.632)

,

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

131

Theorem 2.218. If Re (α) > 0, Re (β) > 0, and s ∈ N, then we have 

1

s

t α−1 (z − t)β−1 eλ(1−t) dt 0

β β+1 β+s−1 ; s , s ,··· , s = B (α, β) × s Fs α+β α+β+1 λ , , · · · , α+β+s−1 ; s , s s

(2.633)

where λ is a constant. Theorem 2.219. If Re (α) > 0, Re (β) > 0, Re (c1 ) > 0, and s ∈ N, then we have 

1

  t α−1 (1 − t)β−1 0 F1 −; c1 ; λ (1 − t)s dt

0

= B (α, β) × s F1+s

β β+1 β+s−1 ; s , s ,··· , s λ α+β α+β+1 α+β+s−1 ; c1 , s , s , · · · , s

(2.634)

,

where λ is a constant. Theorem 2.220. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and s ∈ N, then we have 

1

 a t α−1 (1 − t)β−1 1 − λ (1 − t)s 1 dt

0

= B (α, β) × 1+s Fs

β+s−1 a1 , βs , β+1 ; s ,··· , s λ α+β α+β+1 α+β+s−1 , , · · · , ; s s s

(2.635)

,

where λ is a constant. Theorem 2.221. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, and s ∈ N, then we have 

1

  t α−1 (1 − t)β−1 1 F1 a1 ; c1 ; λ (1 − t)s dt

0

= B (α, β) × 1+s F1+s



β+s−1 a1 , βs , β+1 ; s ,··· , s λ α+β α+β+1 α+β+s−1 c1 , s , s , · · · , ; s

(2.636)

,

where λ is a constant. Theorem 2.222. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, and s ∈ N, then we have

132

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

1

  t α−1 (1 − t)β−1 1 F2 a1 ; c1 , c2 ; λ (1 − t)s dt

0

= B (α, β)



β+s−1 a1 , βs , β+1 ; s ,··· , s λ α+β α+β+1 α+β+s−1 c1 , c2 , s , s , · · · , ; s

× 1+s F2+s

(2.637)

,

where λ is a constant. Theorem 2.223. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and s ∈ N, then we have  1   t α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; λ (1 − t)s dt 0

= B (α, β) × 2+s F1+s



β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s λ α+β α+β+1 α+β+s−1 ; c1 , s , s , · · · , s

(2.638)

,

where λ is a constant.   Theorem 2.224. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, p, q ∈ N0 , s ∈ N, and |z| < 1, then we have 

  t α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t) dt 0   a1 , · · · , ap , β; λ , = B (α, β) × p+1 Fq+1 c1 , · · · , cq , α + β; 1

(2.639)

where λ is a constant. Theorem 2.225. If Re (α) > 0 and Re (β) > 0, then we have 



1

t

α−1

(1 − t)

β−1 λ(1−t)

0

e

dt = B (α, β) × 1 F1

 β; λ , α + β;

(2.640)

where λ is a constant. Theorem 2.226. If Re (α) > 0, Re (β) > 0, and Re (c1 ) > 0, then we have 

1

t α−1 (1 − t)β−1 0 F1 (−; c1 ; λ (1 − t))dt   β; λ , = B (α, β) × 1 F2 c1 , α + β; 0

where λ is a constant.

(2.641)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

133

Theorem 2.227. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then we have 

1

t α−1 (1 − t)β−1 (1 − λ + λt)a1 dt   a1 , β; λ , = B (α, β) × 2 F1 α + β; 0

(2.642)

where λ is a constant. Theorem 2.228. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and Re (c1 ) > 0, then we have 

1

t α−1 (1 − t)β−1 1 F1 (a1 ; c1 ; λ (1 − t)) dt   a1 , β; λ , = B (α, β) × 2 F2 c1 , α + β; 0

(2.643)

where λ is a constant. Theorem 2.229. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, and Re (c2 ) > 0, then we have  1 t α−1 (1 − t)β−1 1 F2 (a1 ; c1 , c2 ; λ (1 − t))dt 0 (2.644)   a1 , β; λ , = B (α, β) × 2 F3 c1 , c2 , α + β; where λ is a constant. Theorem 2.230. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, and Re (c1 ) > 0, then we have  1 t α−1 (1 − t)β−1 2 F1 (a1 , a2 ; c1 ; λ (1 − t))dt 0 (2.645)   a1 , a2 , β; λ , = B (α, β) × 3 F2 c1 , α + β; where λ is a constant. Theorem 2.231. If Re (α) > 0 and Re (β) > 0, then we have 



1

t

α−1

0

where λ is a constant.

β−1 λ(1−t)2

(1 − t)

e

dt = B (α, β) × 2 F2



β β+1 2, 2 ; λ α+β α+β+1 ; 2 , 2

,

(2.646)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 2.232. If Re (α) > 0 and Re (β) > 0, then we have    1 β; 1 . t α−1 (1 − t)β−1 e1−t dt = B (α, β) × 1 F1 α + β; 0 Theorem 2.233. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then we have    1 a1 , β; a1 +α−1 β−1 t dt = B (α, β) × 2 F1 1 . (1 − t) α + β; 0

(2.647)

(2.648)

Theorem 2.234. If Re (α) > 0 and Re (β) > 0, then we have

β β+1  1 2, 2 ; α−1 β−1 (1−t)2 t e dt = B (α, β)2 F2 α+β α+β+1 1 . (2.649) (1 − t) ; 0 2 , 2   Theorem 2.235. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, s ∈ N, and p, q ∈ N0 , then we have  1   (1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t)s dt −1

= B (α, β) 2α+β−1

β+s−1 a1 , · · · , ap , βs , β+1 , · · · , ; s s λ2s , × p+s Fq+s α+β+1 α+β+s−1 c1 , · · · , cq , α+β , , · · · , ; s s s

(2.650)

where λ is a constant. Theorem 2.236. If Re (α) > 0, Re (β) > 0, and s ∈ N, then we have  1 s (1 + t)α−1 (1 − t)β−1 eλ(1−t) dt −1

= B (α, β) 2α+β−1

β β+1 β+s−1 , , · · · , ; s s λ2s , × s Fs α+β s α+β+1 α+β+s−1 , , · · · , ; s s s

(2.651)

where λ is a constant. Theorem 2.237. If Re (α) > 0, Re (β) > 0, Re (c1 ) > 0, and s ∈ N, then we have  1   (1 + t)α−1 (1 − t)β−1 0 F1 −; c1 ; λ (1 − t)s dt −1

= B (α, β) 2α+β−1

β β+1 β+s−1 , , · · · , ; s s s λ2s , × s F1+s α+β+1 α+β+s−1 c1 , α+β , , · · · , ; s s s where λ is a constant.

(2.652)

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

135

Theorem 2.238. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and s ∈ N, then we have 

1 −1

 a (1 + t)α−1 (1 − t)β−1 1 − λ (1 − t)s 1 dt

= B (α, β) 2α+β−1

β+s−1 a1 , βs , β+1 ; s ,··· , s s × 1+s Fs α+β α+β+1 λ2 , , · · · , α+β+s−1 ; s , s s

(2.653)

where λ is a constant. Theorem 2.239. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, and s ∈ N, then we have  1   (1 + t)α−1 (1 − t)β−1 1 F1 a1 ; c1 ; λ (1 − t)s dt −1

= B (α, β) 2α+β−1

β+s−1 a1 , βs , β+1 ; s ,··· , s s × 1+s F1+s λ2 , α+β+1 , · · · , α+β+s−1 ; c1 , α+β s , s s

(2.654)

where λ is a constant. Theorem 2.240. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, and s ∈ N, then we have  1   (1 + t)α−1 (1 − t)β−1 1 F2 a1 ; c1 , c2 ; λ (1 − t)s dt −1

= B (α, β) 2α+β−1

β+s−1 a1 , βs , β+1 ; s ,··· , s s × 1+s F2+s λ2 , α+β+1 c1 , c2 , α+β , · · · , α+β+s−1 ; s , s s

(2.655)

where λ is a constant. Theorem 2.241. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and s ∈ N, then we have  1   (1 + t)α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; λ (1 − t)s dt −1

= B (α, β) 2α+β−1

β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s s × 2+s F1+s λ2 , α+β+1 , · · · , α+β+s−1 ; c1 , α+β s , s s where λ is a constant.

(2.656)

  Theorem 2.242. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, · · · , Re ap > 0, Re (c1 ) > 0, · · · ,   Re cq > 0, and p, q ∈ N0 , then we have

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1 −1

  (1 + t)α−1 (1 − t)β−1 p Fq a1 , · · · , ap ; c1 , · · · , cq ; λ (1 − t) dt 

= B (α, β) 2

α+β−1

× p+1 Fq+1

 a1 , · · · , ap , β; 2λ , c1 , · · · , cq , α + β;

(2.657)

where λ is a constant. Theorem 2.243. If Re (α) > 0 and Re (β) > 0, then we have 

1

−1

 (1 + t)

α−1

(1 − t)

β−1 λ(1−t)

e

dt = B (α, β) 2

α+β−1

× 1 F1

 β; 2λ , α + β;

(2.658)

where λ is a constant. By the above equations, we obtain the following: Theorem 2.244. If Re (α) > 0, Re (β) > 0, and Re (c1 ) > 0, then we have 

1

−1

(1 + t)α−1 (1 − t)β−1 0 F1 (−; c1 ; λ (1 − t))dt

= B (α, β) 2α+β−1 × 1 F2



 β; 2λ , c1 , α + β;

(2.659)

where λ is a constant. For more results on the Clausen hypergeometric series, see Bailey (1935) [60], Dixon (1891) [89], Bateman (1942) [90], Bailey (1935) [91], Bailey (1928) [92], Barnes (1908) [93], Watson (1924) [94], Roy (1987) [65], Joichi and Stanton (1987) [95], Whipple (1926) [96], Agarwal (1963) [97], Andrews (1974) [98], Hill (1907) [99], Koornwinder (2015) [100], and Yakubovich (2004) [101].

2.5.2 The results on the zeros of the new special functions We now consider the results on the zeros of the new special functions without proofs. Theorem 2.245. The hypergeometric supersine via Clausen hypergeometric series is defined as   (2.660) p Supersinq a1 , · · · , ap ; c1 , · · · , cq ; 0 = 0, where an , cn ∈ C and p, q ∈ N0 . Theorem 2.246. The hypergeometric supercosine via Clausen hypergeometric series is defined as   (2.661) p Supercos q a1 , · · · , ap ; c1 , · · · , cq ; 0 = 1, where an , cn ∈ C and p, q ∈ N0 .

Chapter 2 • Hypergeometric, supertrigonometric, and superhyperbolic functions

137

Theorem 2.247. The hypergeometric supertangent via Clausen hypergeometric series is defined as   (2.662) p Supertanq a1 , · · · , ap ; c1 , · · · , cq ; z = 0, where an , cn ∈ C and p, q ∈ N0 . Theorem 2.248. The hypergeometric supersecant via Clausen hypergeometric series is defined as   (2.663) p Supersec q a1 , · · · , ap ; c1 , · · · , cq ; 0 = 1, where an , cn ∈ C and p, q ∈ N0 . Theorem 2.249. The hypergeometric superhyperbolic sine via Clausen hypergeometric series is defined as   (2.664) p Supersinhq a1 , · · · , ap ; c1 , · · · , cq ; 0 = 0, where an , cn ∈ C and p, q ∈ N0 . Theorem 2.250. The hypergeometric superhyperbolic cosine via Clausen hypergeometric series is defined as   (2.665) p Supercoshq a1 , · · · , ap ; c1 , · · · , cq ; 0 = 1, where an , cn ∈ C and p, q ∈ N0 . Theorem 2.251. The hypergeometric superhyperbolic tangent via Clausen hypergeometric series is defined as   (2.666) p Supertanhq a1 , · · · , ap ; c1 , · · · , cq ; 0 = 0, where an , cn ∈ C and p, q ∈ N0 . Theorem 2.252. The hypergeometric superhyperbolic secant via Clausen hypergeometric series is defined as   (2.667) p Supersechq a1 , · · · , ap ; c1 , · · · , cq ; 0 = 1, where an , cn ∈ C and p, q ∈ N0 . The results are easily obtained by the definitions of the special functions.

3 Hypergeometric supertrigonometric and superhyperbolic functions via Gauss hypergeometric series 3.1 Gauss hypergeometric series In this section, we introduce the history, definition, convergence, properties, and theorems for the Gauss hypergeometric series.

3.1.1 Definition, convergence, and properties for the Gauss hypergeometric series To start with, we give the definitions, convergence, properties, and theorems for the Gauss hypergeometric series. In 1812, Gauss [26] introduced the well-known Gauss hypergeometric series. Definition 3.1. The Gauss hypergeometric series is defined as [26] 2 F1 (a, b; c; z)

=1+ =

a (a + 1) b (b + 1) z2 ab z+ + ··· c c (c + 1) 2

∞  (a)n (b)n zn n=0

(c)n

n!

(3.1)

,

where a, b, c, z ∈ C, n ∈ N0 , and |z| < 1. The Gauss hypergeometric series (3.1) converges for |z| < 1 and diverges for |z| > 1. Theorem 3.1 (Euler theorem). [Euler (1679)] If a, b, c, z ∈ C and |z| < 1, then  (c) 2 F1 (a, b; c; z) =  (b)  (c − b)



1

t b−1 (1 − t)c−b−1 (1 − zt)−a dt,

(3.2)

0

where Re (c) > Re (b) > 0. The result was obtained by Euler in 1679 [102]. An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions https://doi.org/10.1016/B978-0-12-824154-7.00009-X Copyright © 2021 Elsevier Inc. All rights reserved.

139

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Theorem 3.2 (Gauss theorem). [Gauss (1812)] If Re (c − a − b) > 0, Re (c) > Re (a), and Re (c) > Re (b), then 2 F1 (a, b; c; 1) =

∞   (c)  (c − b − a) (a)n (b)n 1 . = (c)n n!  (c − a)  (c − b)

(3.3)

n=0

For |z| < 1 and a, b, c, z ∈ C, the function 2 F1 (a, b; c; z) is analytic except for simple poles at c = 0 and c ∈ Z− . The result was obtained by Gauss in 1812 [26]. Theorem 3.3 (Chu–Vandermonde identity). [Chu (1303); Vandermonde (1772)] 2 F1 (−n, b; c; 1) =

(c − b)n . (c)n

(3.4)

The result was obtained by Chu in 1303 [103] and Vandermonde in 1772 [104]. Theorem 3.4. [Andrews, Askey, and Roy (1999)] If a, b, c, z ∈ C, n ∈ N, and |z| < 1, then 2 F1 (a, b; c; z) =

 (c)  (b)  (c − b)



1

t b−1 (1 − t)c−b−1 1 F0 (a; −; zt)dt,

(3.5)

0

where Re (c) > Re (b) > 0. The result was obtained by Andrews, Askey, and Roy in 1999 [25]. Theorem 3.5. [Andrews, Askey, and Roy (1999)] If Re (c) > Re (b) > 0, |arg (1 − z)| < π, and |z| < 1, then 2 F1 (a, b; c; z) =

 (c)  (d)  (c − d)



1

t d−1 (1 − t)c−d−1 2 F1 (a, b; d, zt)dt.

(3.6)

0

The result was obtained by Andrews, Askey, and Roy in 1999 [25]. Theorem 3.6 (Stirling theorem). [Stirling (1730)] If Re (c) > Re (b) > 0, |arg (1 − z)| < π, and |z| < 1, then  −1 2 F1 1, c − b; c; 2 F1 (1, b; c; z) = (1 − z)

 z . z−1

(3.7)

 z . z−1

(3.8)

The result was obtained by Stirling in 1730 [46]. Theorem 3.7 (Pfaff theorem). [Pfaff (1797)] If Re (c) > Re (b) > 0, |arg (1 − z)| < π and |z| < 1, then  −a F F b; c; z) = − z) (a, (1 2 1 a, c − b; c; 2 1

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

141

The result was obtained by Pfaff in 1797 [105]. Theorem 3.8. [Euler (1794)] If Re (c) > Re (b) > 0, |arg (1 − z)| < π, and |z| < 1, then 2 F1 (a, b; c; z) = (1 − z)

c−a−b

2 F1 (c − a, c − b; c; z).

(3.9)

The result was obtained by Euler in 1794 [106]. Theorem 3.9. [Andrews, Askey, and Roy (1999)] If Re (c) > Re (b) > 0 and |z| < 1, then  (c)  (c − b − a) 2 F1 (a, b; a + b + 1 − c; 1 − z)  (c − a)  (c − b)  (c)  (b + a − c) + (1 − z)b+a−c 2 F1 (c − a, c − b; 1 + c − a − b; z)  (a)  (b)

2 F1 (a, b; c; z) =

(3.10)

and 2 F1 (−n, b; c; z) =

(c − b)n 2 F1 (−n, b; b + 1 − c − n; 1 − z). (c)n

(3.11)

The result was obtained by Andrews, Askey, and Roy in 1999 [25]. Theorem 3.10. [Andrews, Askey, and Roy (1999)] If Re (c) > Re (b) > 0 and |z| < 1, then + b + 1 − c; 1 − z)  (1 − c)  (a + b + 1 − c) = 2 F1 (a, b; c; z)  (a + 1 − c)  (b + 1 − c)  (c − 1)  (b + a + 1 − c) 1−c z 2 F1 (a + 1 − c, b + 1 − c; 1 − c; z). +  (a)  (b)

2 F1 (a, b; a

(3.12)

The result was obtained by Andrews, Askey, and Roy in 1999 [25]. Theorem 3.11. [Whittaker and Watson (1902)] If Re (c) > Re (b) > 0 and |z| < 1, then 2 F1 (a, b; c; z)

   (c)  (b − a) 1 −a = (−z) 2 F1 a, a − c + 1; a − b + 1;  (c − a)  (b) z   1  (c)  (a − b) −b . + (−z) 2 F1 b, b − c + 1; b − a + 1;  (c − b)  (a) z

(3.13)

The result was obtained by Whittaker and Watson in 1902 [19]. Theorem 3.12. [Whittaker and Watson (1902)] If Re (c) > Re (b) > 0 and |z| < 1, then 2 F1 (a, b + 1; c; z) − 2 F1 (a, b; c; z) =

az 2 F1 (a+, b + 1; c + 1; z). c

(3.14)

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The result was obtained by Whittaker and Watson in 1902 [19]. Theorem 3.13 (Bateman theorem). [Bateman (1909)] If Re (c) > Re (μ) > 0 and |z| < 1, then   (c + μ) z1−(c+μ) z c−1 t (z − t)μ−1 2 F1 (a, b; c, t)dt. 2 F1 (a, b; c + μ; z) =  (c)  (μ) 0

(3.15)

The result was obtained by Bateman in 1909 [107]. Theorem 3.14. [Bateman (1909)] If Re (c) > Re (μ) > 0 and |z| < 1, then  (c + μ) 2 F1 (a, b; c + μ; z) =  (c)  (μ)



1

t c−1 (1 − t)μ−1 2 F1 (a, b; c, zt)dt.

(3.16)

0

This is the particular case of the Bateman theorem [107]. Theorem 3.15 (Koshliakov theorem). [Koshliakov (1926)] If Re (c) > Re (μ) > 0 and |z| < 1, then  1  (c + μ) t c−1 (1 − t)μ−1 1 F1 (a, b; c, zt)dt. 1 F1 (a, b; c + μ; z) =  (c)  (μ) 0

(3.17)

This result is well known as the Koshliakov theorem [108]. Theorem 3.16. [Andrews, Askey, and Roy (1999)] If Re (c) > Re (b) > 0 and |z| < 1, then 2 F1 (a, b; c; z) = (1 − z) 2 F1 (a

+ 1, b; c; z) +

(c − b) z 2 F1 (a + 1, b; c + 1; z). c

(3.18)

The result was obtained by Andrews, Askey, and Roy in 1999 [25]. Theorem 3.17. [Andrews, Askey, and Roy (1999)] If Re (c) > Re (b) > 0 and |z| < 1, then 2 F1 (a, b; c; z) = 2 F1 (a, b + 1; c + 1; z) −

a (c − b) z 2 F1 (a + 1, b + 1; c + 2; z). c (c + 1)

(3.19)

The result was obtained by Andrews, Askey, and Roy in 1999 [25].

3.1.2 Gauss differential equations In this part, we present the Gauss differential equations via Gauss hypergeometric series. Property 3.1. The hypergeometric series (3.1) is a solution of the Gauss differential equation z (1 − z)

d 2ϕ dϕ − abϕ = 0, + (c − (a + b + 1) z) dz dz2

where a, b, z ∈ C, c ∈ C\Z− 0 , and |z| < 1.

(3.20)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

143

The work was discovered by Whittaker and Watson in 1927 (see [19] p. 283) and reported in [60]. Property 3.2. The hypergeometric series of the form ϕ (z) = 2 F1 (a, b; c; λz) =

∞  (a)n (b)n (λz)n n=0

(c)n

n!

,

(3.21)

is a solution of the Gauss differential equation z (1 − λz) d 2 ϕ c − (a + b + 1) λz dϕ − abϕ = 0, + λ λ dz dz2

(3.22)

where a, b, λ, z ∈ C, c ∈ C\Z− 0 , and |z| < 1. For a, b, c, z ∈ C, and n ∈ N, we have the equation 2 F1 (a, b; c; λz)

=1+ =

ab a (a + 1) b (b + 1) (λz)2 + ··· (λz) + c c (c + 1) 2

∞  (a)n (b)n (λz)n n=0

(c)n

n!

(3.23)

,

which leads to the series representations 2 F1 (a, b; c; −iλz)

=1+ =

ab a (a + 1) b (b + 1) (−iλz)2 + ··· (−iλz) + c c (c + 1) 2

∞  (a)n (b)n (−iλz)n

(c)n

n=0

n!

(3.24)

,

2 F1 (a, b; c; iλz)

=1+ =

ab a (a + 1) b (b + 1) (iλz)2 + ··· (iλz) + c c (c + 1) 2

∞  (a)n (b)n (iλz)n n=0

(c)n

n!

(3.25)

,

and 2 F1 (a, b; c; −λz)

=1+ =

∞  (a)n (b)n (−λz)n n=0

where i =

√ −1.

ab a (a + 1) b (b + 1) (−λz)2 + ··· (−λz) + c c (c + 1) 2 (c)n

n!

,

(3.26)

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Theorem 3.18 (Barnes contour integral). The Barnes contour integral representation for the Gauss hypergeometric series is given as follows [93]: 1  (c) 2 F1 (a, b; c; z) =  (a)  (b) 2πi



i∞

−i∞

 (a + s)  (b + s)  (−s) (−z)s ds,  (c + s)

(3.27)

where |arg (−z)| < π. The result was obtained by Barnes in [93].

3.1.3 Laplace and Mellin transforms for the Gauss hypergeometric series In this section, we introduce the Laplace and Mellin transforms for the Gauss hypergeometric series and present identities by means of the Gauss hypergeometric series. Based on the results from Chapter 2, it is not difficult to derive the following theorems. Theorem 3.19 (Laplace transforms for the Gauss hypergeometric series). Let a, b, c, z, λ ∈ C and t ∈ [0, +∞). Then the Laplace transform of the function t λ−1 2 F1 (a, b; c; zt) is as follows:   L t λ−1 2 F1 (a, b; c; zt)    a, b λ−1 =L t ; zt 2 F1 c     ∞ a, b e−st t λ−1 2 F1 ; zt dt = c 0    (λ) λ, a, b z , = λ 3 F1 ; c s s

(3.28)

where the Laplace transform is defined as  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(3.29)

0

Theorem 3.20 (Mellin transforms for the Gauss hypergeometric series). Let a, b, c, z, λ ∈ C, z = 0, t ∈ [0, +∞), 0 < Re ( ) < Re (a), 0 < Re ( ) < Re (b), and 0 < Re ( ) < Re (c). Then the Mellin transform of the Gauss hypergeometric series 2 F1 (a, b; c; −zt) is as follows: M [2 F1 (a, b; c; −zt)]  ∞ = t  −1 2 F1 (a, b; c; −zt) dt 0

=

 (a −  )  (b −  )  (c)  (ω) − z ,  (a)  (b)  (c −  )

(3.30)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

145

where the Mellin transform is defined as  M [f (t)] = f ( ) =

∞

t  −1 f (t) dt.

(3.31)

0

Note that the results can be derived from Chapter 2.

3.2 Hypergeometric supertrigonometric functions via Gauss hypergeometric series In this section, we present the concepts and theorems for the hypergeometric supertrigonometric functions via Gauss hypergeometric series.

3.2.1 The definitions, properties, and theorems for the hypergeometric supertrigonometric functions via Gauss hypergeometric series By the definition of the Gauss hypergeometric series it is not difficult to present the definitions for the hypergeometric supertrigonometric functions. We now start with the definition of the hypergeometric supersine via Gauss hypergeometric series. Definition 3.2. The hypergeometric supersine via Gauss hypergeometric series is defined as ∞  (a)2n+1 (b)2n+1 (−1)n z2n+1 , (3.32) Supersin b; c; z) = (a, 2 1 (c)2n+1 (2n + 1)! n=0

where a, b, c, z ∈ C. Definition 3.3. The hypergeometric supercosine via Gauss hypergeometric series is defined as ∞  (a)2n (b)2n (−1)n z2n , (3.33) 2 Supercos 1 (a, b; c; z) = (c)2n (2n)! n=0

where a, b, c, z ∈ C. Definition 3.4. The hypergeometric supertangent via Gauss hypergeometric series is defined as 2 Supersin1 (a, b; c; z) , (3.34) 2 Supertan1 (a, b; c; z) = 2 Supercos 1 (a, b; c; z) where a, b, c, z ∈ C. Definition 3.5. The hypergeometric supercotangent via Gauss hypergeometric series is defined as 2 Supercos 1 (a, b; c; z) , (3.35) 2 Supercot 1 (a, b; c; z) = 2 Supersin1 (a, b; c; z) where a, b, c, z ∈ C.

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Definition 3.6. The hypergeometric supersecant via Gauss hypergeometric series is defined as 1 , (3.36) 2 Supersec 1 (a, b; c; z) = Supercos 2 1 (a, b; c; z) where a, b, c, z ∈ C. Definition 3.7. The hypergeometric supercosecant via Gauss hypergeometric series is defined as 1 , (3.37) 2 Supercosec 1 (a, b; c; z) = 2 Supersin1 (a, b; c; z) where a, b, c, z ∈ C. It is not difficult to show that 2 Supersin1 (a, b; c; λz)

1 (2 F1 (a, b; c; iλz) − 2 F1 (a, b; c; −iλz)) 2i ∞  (a)2n+1 (b)2n+1 (−1)n (λz)2n+1 = , (c)2n+1 (2n + 1)!

=

(3.38)

n=0

2 Supercos 1 (a, b; c; λz)

1 (2 F1 (a, b; c; iλz) + 2 F1 (a, b; c; −iλz)) 2 ∞  (a)2n (b)2n (−1)n (λz)2n , = (c)2n (2n)!

=

(3.39)

n=0

2 Supertan1 (a, b; c; λz) = 2 Supercot 1 (a, b; c; λz) = 2 Supersec 1 (a, b; c; λz) =

2 Supersin1 (a, b; c; λz) 2 Supercos 1 (a, b; c; λz) 2 Supercos 1 (a, b; c; λz) 2 Supersin1 (a, b; c; λz)

1 2 Supercos 1 (a, b; c; λz)

,

(3.40)

,

(3.41)

,

(3.42)

and 2 Supercosec 1 (a, b; c; λz) =

1 , 2 Supersin1 (a, b; c; λz)

(3.43)

where a, b, c, λ, z ∈ C. Definition 3.8. The hypergeometric supersine 2 Supersin1 (a, b; c; z), hypergeometric supercosine 2 Supercos 1 (a, b; c; z), hypergeometric supertangent 2 Supertan1 (a, b; c; z), hypergeometric supercotangent 2 Supercot 1 (a, b; c; z), hypergeometric supersecant 2 Supersec 1 (a, b; c; z), and hypergeometric supercosecant 2 Supercosec 1 (a, b; c; z) are called the hypergeometric supertrigonometric functions via Gauss hypergeometric series.

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

Theorem 3.21. If a, b, c, z ∈ C and i =

147

√ −1, then we have

2 F1 (a, b; c; iz) = 2 Supercos 1 (a, b; c; z) + i 2 Supersin1 (a, b; c; z)

(3.44)

2 F1 (a, b; c; −iz) = 2 Supercos 1 (a, b; c; z) − i 2 Supersin1 (a, b; c; z) .

(3.45)

2 F1 (a, b; c; iλz) = 2 Supercos 1 (a, b; c; λz) + i 2 Supersin1 (a, b; c; λz)

(3.46)

2 F1 (a, b; c; −iλz) = 2 Supercos 1 (a, b; c; λz) − i 2 Supersin1 (a, b; c; λz) .

(3.47)

and √ Theorem 3.22. Let a, b, c, λ, z ∈ C, and i = −1. Then we have

and

3.2.2 Fourier–Gauss-type hypergeometric series We now structure the following Fourier–Gauss-type hypergeometric series of the form 2 F1 (a, b; c; z) =

∞ 

φ (κ) (2 F1 (a, b; c; iκz)),

(3.48)

κ=0

which can be represented in the form

 p q a 1 , · · · , a p ; c 1 , · · · , c q ; z = γ +

∞ 





 ϕ (κ) 2 Supercos 1 (a, b; c; kz) + ψ (κ) 2 Supersin1 (a, b; c; kz) ,

(3.49)

κ=1

where γ , φ (κ) =

ϕ (κ) − iψ (κ) , 2

(3.50)

and ϕ (κ) + iψ (κ) (3.51) 2 are the coefficients of the supertrigonometric functions via Gauss hypergeometric series. φ (−κ) =

Theorem 3.23. Let a, b, c ∈ C. Then we have 2 Supersin1 (a, b; c; 0) = 0.

(3.52)

Proof. For a, b, c, λ, z ∈ C, we have the hypergeometric sine 2 Supersin1 (a, b; c; 0) =

1 (2 F1 (a, b; c; 0) − 2 F1 (a, b; c; 0)) = 0. 2i

(3.53)

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Theorem 3.24. Let a, b, c ∈ C. Then we have 2 Supercos 1 (a, b; c; 0) = 1.

(3.54)

Proof. For a, b, c ∈ C, we have the hypergeometric cosine 2 Supercos 1 (a, b; c; 0)

1 (2 F1 (a, b; c; 0) + 2 F1 (a, b; c; 0)) 2 (a)0 (b)0 = (c)0 =

(3.55)

= 1, which satisfies that (a)0 = (b)0 = (c)0 = 1.

Theorem 3.25. Let a, b, c ∈ C. Then we have 2 Supertan1 (a, b; c; 0) = 0.

(3.56)

Proof. For a, b, c ∈ C, we have the hypergeometric tangent 2 Supertan1 (a, b; c; 0) =

2 Supersin1 (a, b; c; 0) 2 Supercos 1 (a, b; c; 0)

= 0,

(3.57)

which is derived from the hypergeometric sine and hypergeometric cosine. Theorem 3.26. Let a, b, c ∈ C. Then we have 2 Supersec 1 (a, b; c; 0) = 1.

(3.58)

Proof. For a, b, c ∈ C, we have the hypergeometric supersecant 2 Supersec 1 (a, b; c; 0) =

1 = 1, 2 Supercos 1 (a, b; c; 0)

(3.59)

which is reduced from the hypergeometric cosine.

3.2.3 The integral transforms for the hypergeometric supertrigonometric functions via Gauss hypergeometric series The Laplace and Mellin transforms for the hypergeometric supersine and supercosine via Gauss hypergeometric series can be written as follows.

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

149

Property 3.3. The Laplace transform of the hypergeometric supersine via Gauss hypergeometric series is given as      (λ) λ, a, b z L t λ−1 2 Supersin1 (a, b; c; zt) = λ × 3 Supersin1 , (3.60) ; c s s where a, b, c, z, λ ∈ C and t ∈ [0, +∞). Property 3.4. The Laplace transform of the hypergeometric supercosine via Gauss hypergeometric series is given as      (λ) λ, a, b z L t λ−1 2 Supercos 1 (a, b; c; zt) = λ × 3 Supercos 1 , (3.61) ; c s s where a, b, c, z, λ ∈ C and t ∈ [0, +∞). Property 3.5. The Mellin transform of the hypergeometric supersine via Gauss hypergeometric series is given as

M 2 Supersin1 (a, b; c; −zt) (3.62)  (iz)−  (a −  )  (b −  )  (c)  (ω)

1 − (−1)− , · = 2i  (a)  (b)  (c −  ) where a, b, c, z, λ ∈ C, z = 0, t ∈ [0, +∞), 0 < Re ( ) < Re (a), 0 < Re ( ) < Re (b), and 0 < Re ( ) < Re (c). Property 3.6. The Mellin transform of the hypergeometric supercosine via Gauss hypergeometric series is given as

M 2 Supercos 1 (a, b; c; −zt) (3.63)  (iz)−  (a −  )  (b −  )  (c)  (ω)

1 + (−1)− , = 2  (a)  (b)  (c −  ) where a, b, c, z, λ ∈ C, z = 0, t ∈ [0, +∞), 0 < Re ( ) < Re (a), 0 < Re ( ) < Re (b), and 0 < Re ( ) < Re (c).

3.3 Hypergeometric superhyperbolic functions via Gauss hypergeometric series In this section, we propose in detail the hypergeometric superhyperbolic functions via Gauss hypergeometric series and their integral transforms.

3.3.1 The definitions, properties, and theorems for the hypergeometric superhyperbolic functions via Gauss hypergeometric series Now we define the hypergeometric superhyperbolic sine via Gauss hypergeometric series.

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Definition 3.9. The hypergeometric superhyperbolic sine via Gauss hypergeometric series is defined as 2 Supersinh1 (a, b; c; z)

1 (2 F1 (a, b; c; λz) − 2 F1 (a, b; c; −z)) 2 ∞  (a)2n+1 (b)2n+1 z2n+1 = , (c)2n+1 (2n + 1)! =

(3.64)

n=0

where a, b, c, z ∈ C. Definition 3.10. The hypergeometric superhyperbolic cosine via Gauss hypergeometric series is defined as 2 Supercosh1 (a, b; c; z)

1 (2 F1 (a, b; c; z) + 2 F1 (a, b; c; −z)) 2 ∞  (a)2n (b)2n z2n , = (c)2n (2n)! =

(3.65)

n=0

where a, b, c, z ∈ C. Definition 3.11. The hypergeometric superhyperbolic tangent via Gauss hypergeometric series is defined as 2 Supertanh1 (a, b; c; z) =

2 Supersinh1 (a, b; c; z) 2 Supercosh1 (a, b; c; z)

,

(3.66)

where a, b, c, z ∈ C. Definition 3.12. The hypergeometric superhyperbolic cotangent via Gauss hypergeometric series is defined as 2 Supercoth1 (a, b; c; z) =

2 Supercosh1 (a, b; c; z) 2 Supersinh1 (a, b; c; z)

,

(3.67)

where a, b, c, z ∈ C. Definition 3.13. The hypergeometric superhyperbolic secant via Gauss hypergeometric series is defined as 2 Supersech1 (a, b; c; z) =

where a, b, c, z ∈ C.

1 , 2 Supercosh1 (a, b; c; z)

(3.68)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

151

Definition 3.14. The hypergeometric superhyperbolic cosecant via Gauss hypergeometric series is defined as 2 Supercosech1 (a, b; c; z) =

1 , Supersinh 2 1 (a, b; c; z)

(3.69)

where a, b, c, λ, z ∈ C. We now derive the relations 2 Supersinh1 (a, b; c; λz)

1 (2 F1 (a, b; c; λz) − 2 F1 (a, b; c; −λz)) 2 ∞  (a)2n+1 (b)2n+1 (λz)2n+1 , = (c)2n+1 (2n + 1)! =

(3.70)

n=0

2 Supercosh1 (a, b; c; λz)

1 (2 F1 (a, b; c; λz) + 2 F1 (a, b; c; −λz)) 2 ∞  (a)2n (b)2n (λz)2n , = (c)2n (2n)! =

(3.71)

n=0

2 Supertanh1 (a, b; c; λz) = 2 Supercoth1 (a, b; c; λz) = 2 Supersech1 (a, b; c; λz) =

2 Supersinh1 (a, b; c; λz)

,

(3.72)

,

(3.73)

1 , 2 Supercosh1 (a, b; c; λz)

(3.74)

2 Supercosh1 (a, b; c; λz) 2 Supercosh1 (a, b; c; λz) 2 Supersinh1 (a, b; c; λz)

and 2 Supercosech1 (a, b; c; λz) =

1 , 2 Supersinh1 (a, b; c; λz)

(3.75)

where a, b, c, λ, z ∈ C. Definition 3.15. The hypergeometric superhyperbolic sine 2 Supersinh1 (a, b; c; z), hypergeometric superhyperbolic cosine 2 Supercosh1 (a, b; c; z), hypergeometric superhyperbolic tangent 2 Supertanh1 (a, b; c; z), hypergeometric superhyperbolic cotangent 2 Supercoth1 (a, b; c; z), hypergeometric superhyperbolic secant 2 Supersech1 (a, b; c; z), and hypergeometric superhyperbolic cosecant 2 Supercosech1 (a, b; c; z) are called the hypergeometric superhyperbolic functions via Gauss hypergeometric series. Theorem 3.27. Let a, b, c, λ, z ∈ C. Then we have 2 F1 (a, b; c; z) = 2 Supercosh1 (a, b; c; z) + 2 Supersinh1 (a, b; c; z)

(3.76)

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and 2 F1 (a, b; c; −z) = 2 Supercosh1 (a, b; c; z) − 2 Supersinh1 (a, b; c; z) .

(3.77)

Moreover, we have 2 F1 (a, b; c; λz) = 2 Supercosh1 (a, b; c; λz) + 2 Supersinh1 (a, b; c; λz)

(3.78)

and 2 F1 (a, b; c; −λz) = 2 Supercosh1 (a, b; c; λz) − 2 Supersinh1 (a, b; c; λz) ,

(3.79)

where a, b, c, λ, z ∈ C. Theorem 3.28. Let a, b, c ∈ C. Then we have 2 Supersinh1 (a, b; c; 0) = 0.

(3.80)

Proof. For a, b, c, λ, z ∈ C, we have the hypergeometric superhyperbolic sine via Gauss hypergeometric series 2 Supersinh1 (a, b; c; 0) =

1 (2 F1 (a, b; c; 0) − 2 F1 (a, b; c; 0)) = 0. 2

(3.81)

Theorem 3.29. Let a, b, c ∈ C. Then we have 2 Supercosh1 (a, b; c; 0) = 1.

(3.82)

Proof. For a, b, c ∈ C, we have the hypergeometric superhyperbolic cosine via Gauss hypergeometric series 2 Supercosh1 (a, b; c; 0) =

1 (2 F1 (a, b; c; 0) + 2 F1 (a, b; c; 0)) = 1. 2

(3.83)

Theorem 3.30. Let a, b, c ∈ C. Then we have 2 Supertanh1 (a, b; c; 0) = 0.

(3.84)

Proof. For a, b, c ∈ C, we have the hypergeometric superhyperbolic tangent via Gauss hypergeometric series 2 Supertanh1 (a, b; c; 0) =

2 Supersinh1 (a, b; c; 0) 2 Supercosh1 (a, b; c; 0)

= 0.

(3.85)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

153

Theorem 3.31. Let a, b, c ∈ C. Then we have 2 Supersec 1 (a, b; c; 0) = 1.

(3.86)

Proof. For a, b, c ∈ C, we have the hypergeometric superhyperbolic secant via Gauss hypergeometric series 2 Supersec 1 (a, b; c; 0) =

1 = 1. 2 Supercos 1 (a, b; c; 0)

(3.87)

3.3.2 The integral transforms for the hypergeometric superhyperbolic functions via Gauss hypergeometric series In this part, we present the integral transforms for the hypergeometric superhyperbolic sine and hypergeometric superhyperbolic cosine via Gauss hypergeometric series. Property 3.7. The Laplace transform of the hypergeometric superhyperbolic sine via Gauss hypergeometric series is given as     (λ)  λ, a, b z L t λ−1 2 Supersinh1 (a, b; c; zt) = λ × 3 Supersinh1 ; , (3.88) c s s where a, b, c, z, λ ∈ C and t ∈ [0, +∞). Property 3.8. The Laplace transform of the hypergeometric superhyperbolic cosine via Gauss hypergeometric series is given as     (λ)  λ, a, b z L t λ−1 2 Supercosh1 (a, b; c; zt) = λ × 3 Supercosh1 ; , (3.89) c s s where a, b, c, z, λ ∈ C and t ∈ [0, +∞). Property 3.9. The Mellin transform of the hypergeometric superhyperbolic sine via Gauss hypergeometric series is given as

M 2 Supersinh1 (a, b; c; −zt) (3.90)  z−  (a −  )  (b −  )  (c)  (ω)

1 − (−1)− , = · 2  (a)  (b)  (c −  ) where a, b, c, z, λ ∈ C, z = 0, t ∈ [0, +∞), 0 < Re ( ) < Re (a), 0 < Re ( ) < Re (b), and 0 < Re ( ) < Re (c). Property 3.10. The Mellin transform of the hypergeometric superhyperbolic cosine via Gauss hypergeometric series is given as

M 2 Supercosh1 (a, b; c; −zt) (3.91)  z−  (a −  )  (b −  )  (c)  (ω)

1 + (−1)− , = 2  (a)  (b)  (c −  )

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

where a, b, c, z, λ ∈ C, z = 0, t ∈ [0, +∞), 0 < Re ( ) < Re (a), 0 < Re ( ) < Re (b), and 0 < Re ( ) < Re (c).

3.4 Some elementary examples for the Gauss hypergeometric series In this section, we present some elementary examples for the Gauss hypergeometric series.

3.4.1 Some results for the Gauss hypergeometric series By the above results it is not difficult to show some elementary examples for the Gauss hypergeometric series (see [45]; [109], p. 15): 0 F0 (−; −; z) = e 0 F0

z

,

 α −; −; zα = ez ,

1 F0 (a; −; z) ∞ n 

=

(a)n

n=0

z n!

= (1 − z)−a ,   1 3 2 ; ; −z z1 F1 2 2 ∞  (−1)n z2n+1 = n! 2n + 1 n=0  z 2 = e−t dt, z ∈ C,

(3.92) (3.93)

(3.94)

(3.95)

0 1 F0 (a; −; 1 − z) ∞ 

=

(a)n

n=0 −a

=z =

1  (a)



(1 − z)n n!

∞

t a−1 e−zt dt,

0

0 F1 (−; c; z) =

 0 F1 (−; c; z) 0 F1 (−; c; z) = 2 F3

(3.96)

∞  1 zn , (c)n n!

(3.97)

n=0

 1 1 (a + b) , (a + b − 1) ; a, b, a + b − 1; 4z , 2 2

(3.98)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions   c 1 z2 a, c − a; c, , , F F F c; z) c; z) = + 1) ; (a; (a; (c 1 1 1 1 2 3 2 2 4 Log (1 + z) = z × 2 F1 (1, 1; 2; −z),   1 1 3 2 , ; ;z , sin−1 x = z × 2 F1 2 2 2   3 z2 , sin x = z × 0 F1 −; ; − 2 4   1 z2 cos x = 0 F1 −; ; − , 2 4   1 3 −1 2 , 1; ; −z , tan x = z × 2 F1 2 2       z  −z −α   |z| < 1,  Re (b) > 0, and |z| < 1. Then we have 2 Supersin1 (a, b; c; z)

 1

  (c) = t b−1 (1 − t)c−b−1 (1 − zt)−a − (1 + zt)−a dt 2i (b)  (c − b) 0  1  (c) = t b−1 (1 − t)c−b−1 1 Supersin0 (a; −; zt)dt.  (b)  (c − b) 0

(3.202)

Proof. For a, b, c, z ∈ C, n ∈ N, and |z| < 1, we have 2 F1 (a, b; c; iz) =

 (c)  (b)  (c − b)

and  (c) 2 F1 (a, b; c; −iz) =  (b)  (c − b)



1 t b−1 (1 − t)c−b−1

0

 0

(1 − izt)a

dt

1 t b−1 (1 − t)c−b−1

(1 + izt)a

dt,

(3.203)

(3.204)

so that 2 Supersin1 (a, b; c; z) ∞  (a)2n+1 (b)2n+1

=

n=0

(c)2n+1

(−1)n z2n+1 , (2n + 1)!

1 (2 F1 (a, b; c; iz) − 2 F1 (a, b; c; −iz)) 2i   1  1

 (c) t b−1 (1 − t)c−b−1 = (1 − zt)−a − (1 + zt)−a dt  (b)  (c − b) 0 2i  1  (c) = t b−1 (1 − t)c−b−1 1 Supersin0 (a; −; zt)dt  (b)  (c − b) 0 =

(3.205)

due to 1 Supersin0 (a; −; zt) =

 1

(1 − izt)−a − (1 + izt)−a , 2i

(3.206)

where Re (c) > Re (b) > 0. Theorem 3.33. Let a, b, c, z ∈ C, n ∈ N, Re (c) > Re (b) > 0, and |z| < 1. Then we have 2 Supercosh1 (a, b; c; z)

 1

  (c) t b−1 (1 − t)c−b−1 (1 − izt)−a + (1 + izt)−a dt 2 (b)  (c − b) 0  1  (c) t b−1 (1 − t)c−b−1 1 Supercos 0 (a; −; zt)dt. =  (b)  (c − b) 0 =

(3.207)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

171

Proof. For a, b, c, z ∈ C, n ∈ N, and |z| < 1, we have 

 (c) 2 F1 (a, b; c; iz) =  (b)  (c − b)

(1 − izt)a

0

and 2 F1 (a, b; c; −iz) =

1 t b−1 (1 − t)c−b−1

 (c)  (b)  (c − b)



dt

1 t b−1 (1 − t)c−b−1

0

(1 + izt)a

dt,

(3.208)

(3.209)

so that 2 Supercos 1 (a, b; c; z) ∞  (a)2n (b)2n (−1)n z2n

=

n=0

(c)2n

(2n)!

1 (2 F1 (a, b; c; iz) + 2 F1 (a, b; c; −iz)) 2   1  1

 (c) = t b−1 (1 − t)c−b−1 (1 − izt)−a + (1 + izt)−a dt  (b)  (c − b) 0 2  1  (c) t b−1 (1 − t)c−b−1 1 Supercos 0 (a; −; zt)dt, =  (b)  (c − b) 0

=

since 1 Supercos 0 (a; −; zt) =

 1

(1 − izt)−a + (1 + izt)−a , 2

(3.210)

where Re (c) > Re (b) > 0. Theorem 3.34. Let a, b, c, z ∈ C, n ∈ N, Re (c) > Re (b) > 0, and |z| < 1. Then we have 2 Supersinh1 (a, b; c; z)

 1

  (c) t b−1 (1 − t)c−b−1 (1 − zt)−a − (1 + zt)−a dt 2 (b)  (c − b) 0  1  (c) = t b−1 (1 − t)c−b−1 1 Supersinh0 (a; −; zt)dt.  (b)  (c − b) 0 =

(3.211)

Proof. For a, b, c, z ∈ C, n ∈ N, and |z| < 1, we have 2 F1 (a, b; c; z) =

 (c)  (b)  (c − b)

and 2 F1 (a, b; c; −z) =

 (c)  (b)  (c − b)



1 t b−1 (1 − t)c−b−1

0



(1 − zt)a

dt

1 t b−1 (1 − t)c−b−1 0

(1 + zt)a

dt,

(3.212)

(3.213)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and thus 2 Supersinh1 (a, b; c; z) ∞  (a)2n+1 (b)2n+1

=

(c)2n+1

n=0

z2n+1 (2n + 1)!

1 (2 F1 (a, b; c; z) − 2 F1 (a, b; c; −z)) 2   1

  (c) b−1 c−b−1 1 −a −a = t dt (1 − zt) − (1 + zt) (1 − t)  (b)  (c − b) 0 2  1  (c) t b−1 (1 − t)c−b−1 1 Supersinh0 (a; −; zt)dt =  (b)  (c − b) 0

=

(3.214)

due to 1 Supersinh0 (a; −; zt) =

 1

(1 − zt)−a − (1 + zt)−a , 2

(3.215)

where Re (c) > Re (b) > 0. Theorem 3.35. Let a, b, c, z ∈ C, n ∈ N, Re (c) > Re (b) > 0, and |z| < 1. Then we have 2 Supercosh1 (a, b; c; z)

 1

  (c) = t b−1 (1 − t)c−b−1 (1 − zt)−a + (1 + zt)−a dt 2 (b)  (c − b) 0  1  (c) t b−1 (1 − t)c−b−1 1 Supercosh0 (a; −; zt)dt. =  (b)  (c − b) 0

(3.216)

Proof. For a, b, c, z ∈ C, n ∈ N, and |z| < 1, we have 2 F1 (a, b; c; z) =

 (c)  (b)  (c − b)

and  (c) 2 F1 (a, b; c; −z) =  (b)  (c − b)



1 t b−1 (1 − t)c−b−1

0



(1 − zt)a

dt

1 t b−1 (1 − t)c−b−1 0

(1 + zt)a

dt,

so that 2 Supercosh1 (a, b; c; z) ∞  (a)2n (b)2n z2n

=

n=0

(c)2n

(2n)!

1 (2 F1 (a, b; c; z) + 2 F1 (a, b; c; −z)) 2   1  1

 (c) = t b−1 (1 − t)c−b−1 (1 − zt)−a + (1 + zt)−a dt  (b)  (c − b) 0 2 =

(3.217)

(3.218)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

=

 (c)  (b)  (c − b)



1

173

t b−1 (1 − t)c−b−1 1 Supercosh0 (a; −; zt)dt

0

due to 1 Supercosh0 (a; −; zt) =

 1

(1 − zt)−a + (1 + zt)−a , 2

(3.219)

where Re (c) > Re (b) > 0. Theorem 3.36. Let z ∈ C\ {0, i, −i}, z ∈ C, n ∈ N, and |z| < 1. Then we have 2 Supersin1 (1, 1; 2; z)



=

1

1 Supersin0 (1; −; zt)dt

0

=

(3.220)

1 (Log (1 − iz) − Log (1 + iz)) . 2z

Proof. For z ∈ C\ {0, i, −i}, z ∈ C, n ∈ N, and |z| < 1, we have  1 F 1; 2; iz) = (1, (1 − izt)−a dt 2 1

(3.221)

0

and



1

2 F1 (1, 1; 2; −iz) =

(1 + izt)−a dt,

(3.222)

0

so that 2 Supersin1 (1, 1; 2; z) ∞  (1)2n+1 (1)2n+1

=

n=0

(2)2n+1

(−1)n z2n+1 (2n + 1)!

1 (Log (1 − iz) − Log (1 + iz)) 2z 1 = (2 F1 (1, 1; 2; iz) − 2 F1 (1, 1; 2; −iz)) 2i   1 1

= (1 − izt)−a − (1 + izt)−a dt 2i 0  1 = 1 Supersin0 (a; −; zt)dt, =

(3.223)

0

where 1 Supersin0 (1; −; zt) =

 1  (1 − izt)−1 − (1 + izt)−1 . 2i

(3.224)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 3.37. Let z ∈ C\ {0, i, −i}, z ∈ C, n ∈ N, and |z| < 1. Then we have 2 Supercosh1 (1, 1; 2; z)



1

=

1 Supercos 0 (1; −; zt) dt

0

=−

(3.225)

1 (Log (1 − iz) + Log (1 + z)) . 2zi

Proof. For z ∈ C\ {0, i, −i}, z ∈ C, n ∈ N, and |z| < 1, we have 

1

2 F1 (1, 1; 2; iz) =

(1 − izt)−a dt

(3.226)

0

and



1

2 F1 (1, 1; 2; −iz) =

(1 + izt)−a dt,

(3.227)

0

so that 2 Supercos 1 (1, 1; 2; z)

1 (Log (1 − iz) + Log (1 + z)) 2zi ∞  (1)2n (1)2n (−1)n z2n

=− =

(2)2n

n=0

(2n)!

1 (2 F1 (1, 1; 2; iz) + 2 F1 (1, 1; 2; −iz)) 2   1 1 = (1 − izt)−1 + (1 + izt)−1 dt 2 0  1 = 1 Supercos 0 (1; −; zt) dt,

=

0

where 1 Supercos 0 (1; −; zt) =

 1 (1 − izt)−1 + (1 + izt)−1 . 2

(3.228)

Theorem 3.38. Let z ∈ C\ {0, i, −i}, z ∈ C, n ∈ N, and |z| < 1. Then we have 2 Supersinh1 (1, 1; 2; z)



=

1

1 Supersinh0 (1; −; zt)dt

0

=

1 (Log (1 + z) − Log (1 − z)) . 2z

(3.229)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

175

Proof. For z ∈ C\ {0, i, −i}, z ∈ C, n ∈ N, and |z| < 1, we have 

1

2 F1 (1, 1; 2; z) =

(1 − zt)−1 dt

(3.230)

0

and



1

2 F1 (1, 1; 2; −z) =

(1 + zt)−1 dt,

(3.231)

0

so that 2 Supersinh1 (1, 1; 2; z) ∞  (1)2n+1 (1)2n+1

=

z2n+1 (2n + 1)!

(2)2n+1

n=0

1 (2 F1 (1, 1; 2; z) − 2 F1 (1, 1; 2; −z)) 2   1 1 = (1 − zt)−1 − (1 + zt)−1 dt 2 0  1 = 1 Supersinh0 (1; −; zt)dt, =

(3.232)

0

where 1 Supersinh0 (1; −; zt) =

 1 (1 − zt)−1 − (1 + zt)−1 . 2

(3.233)

Theorem 3.39. Let z ∈ C\ {0, i, −i}, z ∈ C, n ∈ N, and |z| < 1. Then we have 2 Supercosh1 (a, b; c; z)



1

=

1 Supercosh0 (1; −; zt)dt

0

=−

(3.234)

1 (Log (1 + z) + Log (1 − z)) . 2z

Proof. For z ∈ C\ {0, i, −i}, z ∈ C, n ∈ N, and |z| < 1, we have 

1

2 F1 (1, 1; 2; z) =

(1 − zt)−1 dt

(3.235)

0

and

 2 F1 (1, 1; 2; −z) = 0

1

(1 + zt)−1 dt,

(3.236)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

so that 2 Supercosh1 (1, 1; 2; z) ∞  (1)2n (1)2n z2n

=

n=0

(2)2n

(2n)!

1 (2 F1 (1, 1; 2; z) + 2 F1 (1, 1; 2; −z)) 2   1 1 = (1 − zt)−1 + (1 + zt)−1 dt 2 0  1 = 1 Supercosh0 (1; −; zt)dt,

=

0

where 1 Supercosh0 (1; −; zt) =

 1 (1 − zt)−1 + (1 + zt)−1 . 2

(3.237)

Theorem 3.40. Let a, b, c, z ∈ C, |arg (−z)| < π, Re (c) > Re (b) > 0, and |z| < 1. Then we have 2 Supersin1 (a, b; c; z)

 i∞ 1  (c)  (a + s)  (b + s)  (−s) =  (a)  (b) 2πi −i∞  (c + s)  1 × (1 F0 (−s; −; 1 + iz) − 1 F0 (−s; −; 1 − iz)) ds. 2i

(3.238)

Proof. For a, b, c, z ∈ C, |arg (−z)| < π, Re (c) > Re (b) > 0, and |z| < 1, we have 2 F1 (a, b; c; −iz) =

1  (c)  (a)  (b) 2πi



 (a + s)  (b + s)  (−s) (iz)s ds  (c + s)

(3.239)

 (a + s)  (b + s)  (−s) (−iz)s ds,  (c + s)

(3.240)

i∞ −i∞

and 2 F1 (a, b; c; iz) =

1  (c)  (a)  (b) 2πi



i∞

−i∞

so that 2 Supersin1 (a, b; c; z) ∞  (a)2n+1 (b)2n+1

=

n=0

=

(c)2n+1

(−1)n z2n+1 (2n + 1)!

1 (2 F1 (a, b; c; iz) − 2 F1 (a, b; c; −iz)) 2i

(3.241)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

177

 i∞  (c) 1  (a + s)  (b + s)  (−s)  (a)  (b) 2πi −i∞  (c + s)  1 × (1 F0 (−s; −; 1 + iz) − 1 F0 (−s; −; 1 − iz)) ds, 2i =

where 1 F0 (−a; −; 1 − z) =

∞ 

(−a)n

n=0

(1 − z)n = za . n!

(3.242)

Theorem 3.41. Let a, b, c, z ∈ C, |arg (−z)| < π, Re (c) > Re (b) > 0, and |z| < 1. Then we have 2 Supercos 1 (a, b; c; z)

 i∞ 1  (a + s)  (b + s)  (−s)  (c) =  (a)  (b) 2πi −i∞  (c + s)  1 × (1 F0 (−s; −; 1 + iz) + 1 F0 (−s; −; 1 − iz)) ds. 2

Proof. For a, b, c, z ∈ C, |arg (−z)| < π, Re (c) > Re (b) > 0, and |z| < 1, we have  i∞ 1  (a + s)  (b + s)  (−s)  (c) F b; c; −iz) = (iz)s ds 2 1 (a,  (a)  (b) 2πi −i∞  (c + s) and 2 F1 (a, b; c; iz) =

1  (c)  (a)  (b) 2πi



i∞

−i∞

 (a + s)  (b + s)  (−s) (−iz)s ds,  (c + s)

(3.243)

(3.244)

(3.245)

so that 2 Supercos 1 (a, b; c; z) ∞  (a)2n (b)2n (−1)n z2n

=

n=0

(c)2n

(2n)!

,

1 (2 F1 (a, b; c; iz) + 2 F1 (a, b; c; −iz)) 2  i∞ 1  (a + s)  (b + s)  (−s)  (c) =  (a)  (b) 2πi −i∞  (c + s)  1 × (1 F0 (−s; −; 1 + iz) + 1 F0 (−s; −; 1 − iz)) ds, 2 =

where 1 F0 (−a; −; 1 − z) =

∞  n=0

(−a)n

(1 − z)n = za . n!

(3.246)

(3.247)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 3.42. Let a, b, c, z ∈ C, |arg (−z)| < π, Re (c) > Re (b) > 0, and |z| < 1. Then we have 2 Supersinh1 (a, b; c; z)

 i∞ 1  (c)  (a + s)  (b + s)  (−s) =  (a)  (b) 2πi −i∞  (c + s)  1 × (1 F0 (−s; −; 1 + z) − 1 F0 (−s; −; 1 − z)) ds. 2

(3.248)

Proof. For a, b, c, z ∈ C, |arg (−z)| < π, Re (c) > Re (b) > 0, and |z| < 1, we have 2 F1 (a, b; c; −z) =

1  (c)  (a)  (b) 2πi



i∞

−i∞

 (a + s)  (b + s)  (−s) s z ds  (c + s)

(3.249)

and 2 F1 (a, b; c; z) =

1  (c)  (a)  (b) 2πi



i∞

−i∞

 (a + s)  (b + s)  (−s) (−z)s ds,  (c + s)

(3.250)

which lead to the integral relation 2 Supersin1 (a, b; c; z) ∞  (a)2n+1 (b)2n+1

=

n=0

(c)2n+1

z2n+1 , (2n + 1)!

1 (2 F1 (a, b; c; iz) − 2 F1 (a, b; c; −iz)) 2i  i∞ 1  (a + s)  (b + s)  (−s)  (c) =  (a)  (b) 2πi −i∞  (c + s)  1 × (1 F0 (−s; −; 1 + z) − 1 F0 (−s; −; 1 − z)) ds, 2i =

(3.251)

where 1 F0 (−a; −; 1 − z) =

∞  n=0

(−a)n

(1 − z)n = za . n!

(3.252)

Theorem 3.43. Let a, b, c, z ∈ C, |arg (−z)| < π, Re (c) > Re (b) > 0, and |z| < 1. Then we have 2 Supercosh1 (a, b; c; z)

 i∞ 1  (c)  (a + s)  (b + s)  (−s)  (a)  (b) 2πi −i∞  (c + s)  1 × (1 F0 (−s; −; 1 + z) + 1 F0 (−s; −; 1 − z)) ds. 2 =

(3.253)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

179

Proof. For a, b, c, z ∈ C, |arg (−z)| < π, Re (c) > Re (b) > 0, and |z| < 1, we have 2 F1 (a, b; c; −iz) =

1  (c)  (a)  (b) 2πi



 (a + s)  (b + s)  (−s) (iz)s ds  (c + s)

(3.254)

 (a + s)  (b + s)  (−s) (−iz)s ds,  (c + s)

(3.255)

i∞ −i∞

and 2 F1 (a, b; c; iz) =

1  (c)  (a)  (b) 2πi



i∞

−i∞

so that 2 Supercosh1 (a, b; c; z) ∞  (a)2n (b)2n z2n

=

n=0

(c)2n

(2n)!

,

1 (2 F1 (a, b; c; z) + 2 F1 (a, b; c; −z)) 2  i∞ 1  (c)  (a + s)  (b + s)  (−s) =  (a)  (b) 2πi −i∞  (c + s)  1 × (1 F0 (−s; −; 1 + z) + 1 F0 (−s; −; 1 − z)) ds, 2 =

(3.256)

where 1 F0 (−a; −; 1 − z) =

∞  n=0

(−a)n

(1 − z)n = za . n!

(3.257)

As a direct result, we present the following property. Property 3.19. We have 2 Supersinh1 (a, b; c; z) = −i 2 Supersin1 (a, b; c; iz) ,

(3.258)

2 Supercosh1 (a, b; c; z) = 2 Supercos 1 (a, b; c; iz) ,

(3.259)

2 Supertanh1 (a, b; c; z) = −i 2 Supertan1 (a, b; c; iz) ,

(3.260)

2 Supercoth1 (a, b; c; z) = i 2 Supercot 1 (a, b; c; iz),

(3.261)

2 Supersech1 (a, b; c; z) = 2 Supersec 1 (a, b; c; iz) ,

(3.262)

2 Supercosech1 (a, b; c; λz) = i 2 Supercosec 1 (a, b; c; iz).

(3.263)

and

Moreover, we have 1 Supersinh0 (a; −; λz) = −i 1 Supersin0 (a; −; iλz) ,

(3.264)

180

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

1 Supercosh0 (a; −; λz) = 1 Supercos 0 (a; −; iλz) ,

(3.265)

1 Supertanh0 (a; −; λz) = −i 1 Supertan0 (a; −; −λz) ,

(3.266)

1 Supercoth0 (a; −; λz) = i 1 Supercot 0 (a; −; iλz) ,

(3.267)

1 Supersech0 (a; −; λz) = 1 Supersec 0 (a; −; iλz) ,

(3.268)

1 Supercosech0 (a; −; λz) = i 1 Supercosec 0 (a; −; iλz) .

(3.269)

and

Property 3.20. The hypergeometric series ϕ (z) = 2 F1 (a, b; c; iλz) =

∞  (a)n (b)n (iλz)n

(c)n

n=0

(3.270)

n!

= 2 Supercos 1 (a, b; c; λz) + i 2 Supersin1 (a, b; c; λz) is a solution of the Gauss differential equation z (1 − iλz) d 2 ϕ 1 dϕ − abϕ = 0, + (c − iλ (a + b + 1) z) iλ iλ dz dz2 √ where a, b, λ, z ∈ C, i = −1, c ∈ C\Z− 0 , and |z| < 1.

(3.271)

Property 3.21. The hypergeometric series ϕ (z) = 1 F0 (a; −; λz) =

∞ 

(a)n

n=0

(λz)n n!

(3.272)

= 1 Supercosh0 (a; −; λz) + 1 Supersinh0 (a; −; λz) is a solution of the Gauss differential equation z (1 − λz) d 2 ϕ dϕ = 0, + (a + 1) z 2 λ dz dz where a, λ, z ∈ C and |z| < 1.

(3.273)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

181

Property 3.22. The hypergeometric series ϕ (z) = 1 F0 (a; −; iλz) =

∞  n=0

(a)n

(iλz)n n!

(3.274)

= 1 Supercos 0 (a; −; λz) + i 1 Supersin0 (a; −; λz) is a solution of the Gauss differential equation dϕ z (1 − iλz) d 2 ϕ + (a + 1) z = 0, iλ dz dz2 where a, λ, z ∈ C, i =

√

(3.275)

−1, and |z| < 1.

We give some elementary examples for the Gauss hypergeometric series:  z ez = lim 2 F1 1, b; 1; , b→∞ b   1 z2 cosh z = a→∞ lim 2 F1 a, b; ; , 2 4ab b→∞  z , 1 F1 (a; c; z) = lim 2 F1 a, b; c; b→∞ b and 0 F1 (−; c; z) =

 z  . a, b; c; F lim 2 1 a→∞ ab b→∞

(3.276) (3.277) (3.278)

(3.279)

The results were obtained by Sonine in 1880 [110] and reported by different researchers [14,22,25,68,109]. Similarly, for |z| < 1 and z ∈ C\ {i, −i}, we give the following results. Let us consider the series representations 0 F1 (−; c; z) =

∞  1 zn (c)n n!

(3.280)

n=0

and ∞  1 (−z)n . 0 F1 (−; c; −z) = (c)n n! n=0

At the moment, we evaluate the following equalities:

(3.281)

182

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

0 Supersin1 (−; c; z) 0 F1 (−; c; iz) − 0 F1 (−; c; −iz)

=

2i

 ∞ n  − (−iz)n (iz) 1 1 = 2i n! (c)n

(3.282)

n=0

∞  (−1)n z2n+1 , (c)2n+1 (2n + 1)!

=

n=0

0 Supercos 1 (−; c; z) 0 F1 (−; c; iz) + 0 F1 (−; c; −iz)

=

2

 ∞  1 1 (iz)n + (−iz)n = 2 n! (c)n

(3.283)

n=0

∞  (−1)n z2n = , (c)2n (2n)! n=0

0 Supertan1 (−; c; z)

=

0 Supersin1 (−; c; z) 0 Supercos 1 (−; c; z)

1 0 F1 (−; c; iz) − 0 F1 (−; c; −iz) × i 0 F1 (−; c; iz) + 0 F1 (−; c; −iz) ∞ 1 (iz)n −(−iz)n  1 n=0 (c)n n!

 =  i ∞ 1 (iz)n +(−iz)n =

n=0 (c)n

n!

∞ =

(3.284)

(−1)n z2n+1 n=0 (c)2n+1 (2n+1)! ∞ (−1)n z2n n=0 (c)2n (2n)!

,

0 Supercotan1 (−; c; z)

=

0 Supercos 1 (−; c; z) 0 Supersin1 (−; c; z)

=i×

+0 F1 (−; c; −iz) F c; iz) − (−; 0 1 0 F1 (−; c; −iz)0 F1 (−; c; iz) ∞ 1 (iz)n +(−iz)n  n=0 (c)n

n!

(−1)n z2n n=0 (c)2n (2n)! ∞ (−1)n z2n+1 n=0 (c)2n+1 (2n+1)!

,

=i×  ∞ ∞

=

 n n 1 (iz) −(−iz) n=0 (c)n n!

(3.285)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

183

0 Supersec 1 (−; c; z)

1 0 Supercos 1 (−; c; z) 2 = 0 F1 (−; c; iz) + 0 F1 (−; c; −iz) 2

 = n n ∞ 1 (iz) +(−iz) =

n=0 (c)n

n!

1

= ∞

(3.286)

(−1)n z2n n=0 (c)2n (2n)!

,

0 Supercosec 1 (−; c; z)

1 0 Supersin1 (−; c; z) 2i = F c; iz) − (−; 0 1 0 F1 (−; c; −iz) 2i

 = n n ∞ 1 (iz) −(−iz) =

n=0 (c)n

n!

1

= ∞

(3.287)

(−1)n z2n+1 n=0 (c)2n+1 (2n+1)!

,

0 Supersinh1 (−; c; z)

=

0 F1 (−; c; z) − 0 F1 (−; c; −z)

2

n  ∞  1 1 z − (−z)n = 2 n! (c)n

(3.288)

n=0

=

∞  n=0

z2n+1 , (c)2n+1 (2n + 1)! 1

0 Supercosh1 (−; c; z)

=

0 F1 (−; c; z) + 0 F1 (−; c; −z)

2

n  ∞  1 1 z + (−z)n = 2 n! (c)n n=0

=

∞  n=0

1 z2n , (c)2n (2n)!

(3.289)

184

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

0 Supertanh1 (−; c; z)

= = =

0 Supersinh1 (−; c; z) 0 Supercosh1 (−; c; z) 0 F1 (−; c; z) − 0 F1 (−; c; −z) 0 F1 (−; c; z) + 0 F1 (−; c; −z) ∞ 1 zn −(−z)n  n=0 (c)n n! ∞ 1 zn +(−z)n  n=0 (a)n n!

(3.290)

∞ =

1 z2n+1 n=0 (c)2n+1 (2n+1)! ∞ 1 z2n n=0 (c)2n (2n)!

,

0 Supercoth1 (−; c; z)

= = = =

0 Supercosh1 (−; c; z) 0 Supersinh1 (−; c; z) 0 F1 (−; c; z) + 0 F1 (−; c; −z) 0 F1 (−; c; z) − 0 F1 (−; c; −z) ∞ 1 zn +(−z)n  n=0 (c)n n!  ∞ (a)n n n n=0 n! z − (−z) ∞ 1 z2n n=0 (c)2n (2n)! , ∞ 1 z2n+1 n=0 (c)2n+1 (2n+1)!

(3.291)

0 Supersech1 (−; c; z)

1 0 Supercosh1 (−; c; z) 2 = F c; z) + (−; 0 1 0 F1 (−; c; −z) 2

 = n n ∞ 1 z +(−z)

=

n=0 (c)n

= ∞

(3.292)

n!

1

1 z2n n=0 (c)2n (2n)!

,

and 1 Supercosech0 (a; −; z)

1 Supersinh 1 0 (a; −; z) 2 = 0 F1 (−; c; z) − 0 F1 (−; c; −z)

=

(3.293)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

185

2

= ∞

 n n 1 z −(−z) n=0 (c)n n!

1

= ∞

z2n+1 n=0 (c)2n+1 (2n+1)! 1

.

Property 3.23. The hypergeometric series ϕ (z) = 0 F1 (−; c; λz)

(3.294)

∞  1 (λz)n = (c)n n! n=0

is a solution of the Gauss differential equation   2  dϕ 1 d ϕ c z + −z − z = 0, λ λ dz dz2

(3.295)

where λ, z ∈ C, c ∈ C\Z− 0 , and |z| < 1. Property 3.24. The hypergeometric series ϕ (z) = 0 F1 (−; c; iλz) =

∞  1 (iλz)n (c)n n!

(3.296)

n=0

is a solution of the Gauss differential equation  2   dϕ d ϕ c 1 −z − z = 0, z + iλ iλ dz dz2

(3.297)

where λ, z ∈ C, c ∈ C\Z− 0 , and |z| < 1. Theorem 3.44. Let z ∈ C, c ∈ C\Z− 0 , and |z| < 1. Then we have 0 F1 (−; c; iz) = 0 Supersin1 (−; c; z) + i 0 Supercos 1 (−; c; z)

(3.298)

0 F1 (−; c; z) = 0 Supersinh1 (−; c; z) + 0 Supercosh1 (−; c; z).

(3.299)

and

Theorem 3.45. Let z ∈ C, c

∈ C\Z− 0 , and |z| < 1. Then we have

0 Supersin1 (−; c; z) =

0 F1 (−; c; iz) − 0 F1 (−; c; −iz)

(3.300)

2i

and 0 Supercos 1 (−; c; z) =

0 F1 (−; c; iz) + 0 F1 (−; c; −iz)

2

.

(3.301)

186

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Generally, we have 0 F1 (−; c; iλz) = 0 Supersin1 (−; c; λz) + i 0 Supercos 1 (−; c; λz)

(3.302)

0 F1 (−; c; λz) = 0 Supersinh1 (−; c; λz) + 0 Supercosh1 (−; c; λz),

(3.303)

and

so that 0 Supersin1 (−; c; λz) =

0 F1 (−; c; iλz) − 0 F1 (−; c; −iλz)

2i

(3.304)

and 0 Supercos 1 (−; c; λz) =

0 F1 (−; c; iλz) + 0 F1 (−; c; −iλz)

2

,

(3.305)

where λ, z ∈ C, c ∈ C\Z− 0 , and |z| < 1. Moreover, we present 0 Supersinh1 (−; c; z) = −i 0 Supersin1 (−; c; iz) ,

(3.306)

0 Supercosh1 (−; c; z) = 0 Supercos 1 (−; c; iz) ,

(3.307)

0 Supertanh1 (−; c; z) = −i 0 Supertan1 (−; c; −z) ,

(3.308)

0 Supercoth1 (−; c; z) = i 0 Supercot 1 (−; c; iz) ,

(3.309)

0 Supersech1 (a; −; z) = 0 Supersec 1 (a; −; iz) ,

(3.310)

0 Supercosech1 (a; −; z) = i 0 Supercosec 1 (a; −; iz) .

(3.311)

and

Theorem 3.46. If |z| < 1 and a, b, c, μ ∈ C, then 2 Supersin1 (a, b; c + μ; z)

 (c + μ) =  (c)  (μ)



1

t c−1 (1 − t)μ−1 2 Supersin1 (a, b; c; zt)dt

(3.312)

0

and 2 Supercos 1 (a, b; c + μ; z)

 (c + μ) =  (c)  (μ)



1

t c−1 (1 − t)μ−1 2 Supercos 1 (a, b; c; zt)dt.

(3.313)

0

Proof. In terms of the integral relations 2 F1 (a, b; c + μ; iz) =

 (c + μ)  (c)  (μ)

 0

1

t c−1 (1 − t)μ−1 2 F1 (a, b; c, izt)dt

(3.314)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

187

and  (c + μ) 2 F1 (a, b; c + μ; −iz) =  (c)  (μ)



1

t c−1 (1 − t)μ−1 2 F1 (a, b; c, −izt)dt,

(3.315)

0

we have 2 Supersin1 (a, b; c + μ; z)

1 (2 F1 (a, b; c + μ; iz) − 2 F1 (a, b; c + μ; −iz)) 2i    1 1  (c + μ) c−1 μ−1 t = (1 − t) (2 F1 (a, b; c, izt) − 2 F1 (a, b; c, −izt)) dt  (c)  (μ) 0 2i  1  (c + μ) = t c−1 (1 − t)μ−1 2 Supersin1 (a, b; c; zt)dt  (c)  (μ) 0

=

(3.316)

and 2 Supercos 1 (a, b; c + μ; z)

1 (2 F1 (a, b; c + μ; iz) + 2 F1 (a, b; c + μ; −iz)) 2    1 1  (c + μ) = t c−1 (1 − t)μ−1 (2 F1 (a, b; c, izt) + 2 F1 (a, b; c, −izt)) dt  (c)  (μ) 0 2  1  (c + μ) = t c−1 (1 − t)μ−1 2 Supercos 1 (a, b; c; zt)dt,  (c)  (μ) 0 =

(3.317)

which are derived from 1 (2 F1 (a, b; c; iz) − 2 F1 (a, b; c; −iz)) , 2i 1 2 Supercos 1 (a, b; c; z) = (2 F1 (a, b; c; iz) + 2 F1 (a, b; c; −iz)) , 2 1 (2 F1 (a, b; c + μ; iz) − 2 F1 (a, b; c + μ; −iz)) , 2 Supersin1 (a, b; c + μ; z) = 2i

(3.319)

1 (2 F1 (a, b; c + μ; iz) + 2 F1 (a, b; c + μ; −iz)) . 2

(3.321)

2 Supersin1 (a, b; c; z) =

(3.318)

(3.320)

and 2 Supercos 1 (a, b; c + μ; z) =

Theorem 3.47. If |z| < 1 and a, b, c, μ ∈ C, then 2 Supersinh1 (a, b; c + μ; z)

=

 (c + μ)  (c)  (μ)



0

1

t c−1 (1 − t)μ−1 2 Supersinh1 (a, b; c; zt)dt

(3.322)

188

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and 2 Supercosh1 (a, b; c + μ; z)

=

 (c + μ)  (c)  (μ)



1

t c−1 (1 − t)μ−1 2 Supercosh1 (a, b; c; zt)dt.

(3.323)

0

Proof. By means of the integral relations 2 F1 (a, b; c + μ; z) =

 (c + μ)  (c)  (μ)



1

t c−1 (1 − t)μ−1 2 F1 (a, b; c, zt)dt

(3.324)

t c−1 (1 − t)μ−1 2 F1 (a, b; c, −zt)dt,

(3.325)

0

and 2 F1 (a, b; c + μ; −z) =

 (c + μ)  (c)  (μ)



1

0

we have 2 Supersinh1 (a, b; c + μ; z)

1 (2 F1 (a, b; c + μ; z) − 2 F1 (a, b; c + μ; −z)) 2    1  (c + μ) c−1 μ−1 1 = t (1 − t) (2 F1 (a, b; c, zt) − 2 F1 (a, b; c, −zt)) dt  (c)  (μ) 0 2  1  (c + μ) = t c−1 (1 − t)μ−1 2 Supersinh1 (a, b; c; zt)dt  (c)  (μ) 0 =

(3.326)

and 2 Supercosh1 (a, b; c + μ; z)

1 (2 F1 (a, b; c + μ; z) + 2 F1 (a, b; c + μ; −z)) 2    1  (c + μ) c−1 μ−1 1 = t (1 − t) (2 F1 (a, b; c, zt) + 2 F1 (a, b; c, −zt)) dt  (c)  (μ) 0 2  1  (c + μ) = t c−1 (1 − t)μ−1 2 Supercosh1 (a, b; c; zt)dt  (c)  (μ) 0 =

(3.327)

since 1 (2 F1 (a, b; c; iz) − 2 F1 (a, b; c; −z)) , 2 1 2 Supercosh1 (a, b; c; z) = (2 F1 (a, b; c; z) + 2 F1 (a, b; c; −z)) , 2 1 (2 F1 (a, b; c + μ; z) − 2 F1 (a, b; c + μ; −z)) , 2 Supersinh1 (a, b; c + μ; z) = 2i 2 Supersinh1 (a, b; c; z) =

(3.328) (3.329) (3.330)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

189

and 2 Supercosh1 (a, b; c + μ; z) =

1 (2 F1 (a, b; c + μ; z) + 2 F1 (a, b; c + μ; −z)) . 2

Theorem 3.48. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then  z   1 t α−1 (z − t)β−1 1 Supersin0 a1 ; −; λt 2 dt B (α, β) 0   a1 , α2 , α+1 2 ; α+β−1 2 =z × 3 Supersin2 α+β α+β+1 λz , ; 2 , 2  z   1 t α−1 (z − t)β−1 1 Supercos 0 a1 ; −; λt 2 dt B (α, β) 0   a1 , α2 , α+1 2 ; α+β−1 2 =z × 3 Supercos 2 α+β α+β+1 λz , ; 2 , 2  z   1 t α−1 (z − t)β−1 1 Supersinh0 a1 ; −; λt 2 dt B (α, β) 0   a1 , α2 , α+1 2 ; α+β−1 2 =z × 3 Supersinh2 α+β α+β+1 λz , ; 2 , 2

(3.331)

(3.332)

(3.333)

(3.334)

and 

  t α−1 (z − t)β−1 1 Supercosh0 a1 ; −; λt 2 dt 0   a1 , α2 , α+1 2 ; α+β−1 2 =z × 3 Supercosh2 α+β α+β+1 λz , ; 2 , 2

1 B (α, β)

z

(3.335)

where λ is a constant. Proof. Using the integral representations  z   1 t α−1 (z − t)β−1 1 F0 a1 ; −; iλt 2 dt B (α, β) 0   a1 , α2 , α+1 2 ; α+β−1 2 =z × 3 F2 α+β α+β+1 iλz , ; 2 , 2  z   1 t α−1 (z − t)β−1 1 F0 a1 ; −; −iλt 2 dt B (α, β) 0   a1 , α2 , α+1 2 ; α+β−1 2 =z × 3 F2 α+β α+β+1 − iλz , ; 2 , 2

(3.336)

(3.337)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

  t α−1 (z − t)β−1 1 F0 a1 ; −; λt 2 dt 0   α α+1 , , ; a 1 2 2 = zα+β−1 × 3 F2 α+β α+β+1 λz2 , ; 2 , 2  z   1 t α−1 (z − t)β−1 1 F0 a1 ; −; −λt 2 dt B (α, β) 0   α α+1 , , ; a 1 2 2 = zα+β−1 × 3 F2 α+β α+β+1 − λz2 , ; 2 , 2 1 B (α, β)

z

(3.338)

(3.339)

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, we obtain the results. Theorem 3.49. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and |z| < 1, then 

z

t α−1 (z − t)β−1 2 Supersin1 (a1 , a2 ; c1 ; λ (z − t))dt

0

= B (α, β) zα+β−1   a1 , a2 , β; λz , × 3 Supersin2 c1 , α + β;  z t α−1 (z − t)β−1 2 Supercos 1 (a1 , a2 ; c1 ; λ (z − t))dt

(3.340)

0

= B (α, β) zα+β−1   a1 , a2 , β; λz , × 3 Supercos 2 c1 , α + β;  z t α−1 (z − t)β−1 2 Supersinh1 (a1 , a2 ; c1 ; λ (z − t))dt

(3.341)

0

= B (α, β) zα+β−1   a1 , a2 , β; λz , × 3 Supersinh2 c1 , α + β;

(3.342)

and 

z

t α−1 (z − t)β−1 2 Supercosh1 (a1 , a2 ; c1 ; λ (z − t))dt

0

= B (α, β) zα+β−1   a1 , a2 , β; λz , × 3 Supercosh2 c1 , α + β; where λ is a constant.

(3.343)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

191

Proof. In view of 

z

t α−1 (z − t)β−1 2 F1 (a1 , a2 ; c1 ; iλ (z − t))dt   a1 , a2 , β; iλz , = B (α, β) zα+β−1 × 3 F2 c1 , α + β;  z t α−1 (z − t)β−1 2 F1 (a1 , a2 ; c1 ; −iλ (z − t))dt 0   a1 , a2 , β; − iλz , = B (α, β) × zα+β−1 3 F2 c1 , α + β;  z t α−1 (z − t)β−1 2 F1 (a1 , a2 ; c1 ; λ (z − t))dt 0   a1 , a2 , β; λz , = B (α, β) zα+β−1 × 3 F2 c1 , α + β; 0

(3.344)

(3.345)

(3.346)

and 

z

t α−1 (z − t)β−1 2 F1 (a1 , a2 ; c1 ; −λ (z − t))dt 0   a1 , a2 , β; α+β−1 − λz , × 3 F2 = B (α, β) z c1 , α + β;

(3.347)

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and |z| < 1, we obtain the results. Theorem 3.50. If Re (α) > 0, Re (β) > 0, and |z| < 1, then 

  t α−1 (z − t)β−1 sin λ (z − t)2 dt = B (α, β) zα+β−1 0  β β+1  2, 2 ; 2 × 2 Supersin2 α+β α+β+1 λz , ; 2 , 2  z   t α−1 (z − t)β−1 cos λ (z − t)2 dt = B (α, β) zα+β−1 0  β β+1  2, 2 ; 2 × 2 Supercos 2 α+β α+β+1 λz , ; 2 , 2  z   t α−1 (z − t)β−1 sinh λ (z − t)2 dt = B (α, β) zα+β−1 0  β β+1  2, 2 ; 2 × 2 Supersinh2 α+β α+β+1 λz , ; 2 , 2 z

(3.348)

(3.349)

(3.350)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and 

  t α−1 (z − t)β−1 cosh λ (z − t)2 dt = B (α, β) zα+β−1 0  β β+1  2, 2 ; 2 × 2 Supercosh2 α+β α+β+1 λz , ; 2 , 2 z

(3.351)

where λ is a constant. Proof. Use the integral expressions  z 2 t α−1 (z − t)β−1 eiλ(z−t) dt = B (α, β) zα+β−1 0  β β+1  2, 2 ; 2 × 2 F2 α+β α+β+1 iλz , ; 2 , 2  z 2 t α−1 (z − t)β−1 e−iλ(z−t) dt = B (α, β) zα+β−1 0  β β+1  2, 2 ; 2 × 2 F2 α+β α+β+1 − iλz , ; 2 , 2  z 2 t α−1 (z − t)β−1 eλ(z−t) dt = B (α, β) zα+β−1 0  β β+1  2, 2 ; 2 × 2 F2 α+β α+β+1 λz , ; 2 , 2

(3.352)

(3.353)

(3.354)

and 

z

t α−1 (z − t)β−1 e−λ(z−t) dt = B (α, β) zα+β−1 0  β β+1  2, 2 ; 2 × 2 F2 α+β α+β+1 − λz . ; 2 , 2 2

(3.355)

Theorem 3.51. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, s ∈ N, and |z| < 1, then  z

 t α−1 (z − t)β−1 1 Supersin2 a1 ; c1 , c2 ; λ (z − t)s dt 0

= B (α, β) zα+β−1 × 1+s Supersin2+s



β+s−1 a1 , βs , β+1 ; s ,··· , s

α+β+1 ,··· c1 , c2 , α+β s , s

,

α+β+s−1 ; s

 λzs ,

(3.356)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions 

193

 t α−1 (z − t)β−1 1 Supercos 2 a1 ; c1 , c2 ; λ (z − t)s dt

z

0

= B (α, β) zα+β−1



× 1+s Supercos 2+s 

z

β+s−1 a1 , βs , β+1 ; s ,··· , s λzs α+β α+β+1 α+β+s−1 c1 , c2 , s , s , · · · , ; s

(3.357)

 ,

 t α−1 (z − t)β−1 1 Supersinh2 a1 ; c1 , c2 ; λ (z − t)s dt

0

= B (α, β) zα+β−1 × 1+s Supersinh2+s and



z



β+s−1 a1 , βs , β+1 ; s ,··· , s

α+β+1 c1 , c2 , α+β ,··· s , s

,

α+β+s−1 ; s



(3.358)

λzs ,

 t α−1 (z − t)β−1 1 Supercosh2 a1 ; c1 , c2 ; λ (z − t)s dt

0

= B (α, β) zα+β−1 × 1+s Supercosh2+s



β+s−1 a1 , βs , β+1 ; s ,··· , s α+β+1 c1 , c2 , α+β , · · · , α+β+s−1 ; s , s s



(3.359)

λzs ,

where λ is a constant. Proof. We have the integral relations  z

 t α−1 (z − t)β−1 1 F2 a1 ; c1 , c2 ; iλ (z − t)s dt 0



= B (α, β) zα+β−1   β+s−1 a1 , βs , β+1 ; s ,··· , s iλzs , × 1+s F2+s α+β+1 α+β+s−1 , , · · · , ; c1 , c2 , α+β s s s z

(3.360)

 t α−1 (z − t)β−1 1 F2 a1 ; c1 , c2 ; −iλ (z − t)s dt

0

= B (α, β) zα+β−1   β+s−1 a1 , βs , β+1 ; s ,··· , s s × 1+s F2+s − iλz , α+β+1 , · · · , α+β+s−1 ; c1 , c2 , α+β s , s s  z

 t α−1 (z − t)β−1 1 F2 a1 ; c1 , c2 ; λ (z − t)s dt

(3.361)

0

= B (α, β) zα+β−1   β+s−1 a1 , βs , β+1 ; s ,··· , s s λz , × 1+s F2+s α+β+1 , · · · , α+β+s−1 ; c1 , c2 , α+β s , s s

(3.362)

194

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

z

 t α−1 (z − t)β−1 1 F2 a1 ; c1 , c2 ; −λ (z − t)s dt

0

= B (α, β) zα+β−1   β+s−1 a1 , βs , β+1 ; s ,··· , s s − λz , × 1+s F2+s α+β+1 c1 , c2 , α+β , · · · , α+β+s−1 ; s , s s

(3.363)

which lead to the results, where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, s ∈ N, and |z| < 1. Theorem 3.52. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, s ∈ N, and |z| < 1, then 

z

 t α−1 (z − t)β−1 2 Supersin1 a1 , a2 ; c1 ; λ (z − t)s dt

0

= B (α, β) zα+β−1



× 2+s Supersin1+s 

z

β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s

α+β+1 c1 , α+β , · · · , α+β+s−1 ; s , s s

(3.364)

 λzs ,

 t α−1 (z − t)β−1 2 Supercos 1 a1 , a2 ; c1 ; λ (z − t)s dt

0

= B (α, β) zα+β−1



× 2+s Supercos 1+s 

z

β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s λzs α+β α+β+1 α+β+s−1 ; c1 , s , s , · · · , s

(3.365)

 ,

 t α−1 (z − t)β−1 2 Supersinh1 a1 , a2 ; c1 ; λ (z − t)s dt

0

= B (α, β) zα+β−1 × 2+s Supersinh1+s



β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s λzs α+β α+β+1 α+β+s−1 ; c1 , s , s , · · · , s

(3.366)

 ,

and 

z

 t α−1 (z − t)β−1 2 Supercosh1 a1 , a2 ; c1 ; λ (z − t)s dt

0

= B (α, β) zα+β−1 × 2+s Supercosh1+s where λ is a constant.



β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s λzs α+β α+β+1 α+β+s−1 c1 , s , s , · · · , ; s

(3.367)

 ,

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

195

Proof. Using the integral expressions  z

 t α−1 (z − t)β−1 2 F1 a1 , a2 ; c1 ; iλ (z − t)s dt 0



= B (α, β) zα+β−1   β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s s × 2+s F1+s iλz , α+β+1 c1 , α+β , · · · , α+β+s−1 ; s , s s z

(3.368)

 t α−1 (z − t)β−1 2 F1 a1 , a2 ; c1 ; −iλ (z − t)s dt

0

= B (α, β) zα+β−1   β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s s × 2+s F1+s − iλz , α+β+1 c1 , α+β , · · · , α+β+s−1 ; s , s s  z

 t α−1 (z − t)β−1 2 F1 a1 , a2 ; c1 ; λ (z − t)s dt

(3.369)

0

= B (α, β) zα+β−1   β+s−1 a1 , a2 , βs , β+1 , · · · , ; s s λzs , × 2+s F1+s α+β+1 α+β+s−1 c1 , α+β , , · · · , ; s s s

(3.370)

and 

z

 t α−1 (z − t)β−1 2 F1 a1 , a2 ; c1 ; −λ (z − t)s dt

0

= B (α, β) zα+β−1   β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s s × 2+s F1+s − λz , α+β+1 , · · · , α+β+s−1 ; c1 , α+β s , s s

(3.371)

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, s ∈ N, and |z| < 1, we get the results. Theorem 3.53. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then  z 1 t α−1 (z − t)β−1 1 Supersin0 (a1 ; −; λt)dt = zα+β−1 B (α, β) 0   a1 , α; × 2 Supersin1 λz , α + β;  z 1 t α−1 (z − t)β−1 1 Supercos 0 (a1 ; −; λt)dt = zα+β−1 B (α, β) 0   a1 , α; λz , × 2 Supercos 1 α + β;

(3.372)

(3.373)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

1 B (α, β)



z

t α−1 (z − t)β−1 1 Supersinh0 (a1 ; −; λt)dt = zα+β−1   a1 , α; × 2 Supersinh1 λz , α + β;

and 1 B (α, β)

0



(3.374)

z

t α−1 (z − t)β−1 1 Supercosh0 (a1 ; −; λt)dt = zα+β−1 0   a1 , α; × 2 Supercosh1 λz , α + β;

(3.375)

where λ is a constant. Proof. In view of the integral relations  z 1 t α−1 (z − t)β−1 1 F0 (a1 ; −; iλt)dt B (α, β) 0   a1 , α; iλz , = zα+β−1 × 2 F1 α + β;  z 1 t α−1 (z − t)β−1 1 F0 (a1 ; −; −iλt)dt B (α, β) 0   a1 , α; − iλz , = zα+β−1 × 2 F1 α + β;  z 1 t α−1 (z − t)β−1 1 F0 (a1 ; −; λt)dt B (α, β) 0   a1 , α; λz , = zα+β−1 × 2 F1 α + β; and 1 B (α, β)



(3.376)

(3.377)

(3.378)

z

t α−1 (z − t)β−1 1 F0 (a1 ; −; −λt)dt   a1 , α; α+β−1 × 2 F1 − λz , =z α + β; 0

(3.379)

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, we obtain the results. Theorem 3.54. If Re (α) > 0, Re (β) > 0, and |z| < 1, then  z t α−1 (z − t)β−1 sin (λt (z − t)) dt = B (α, β) zα+β−1 0   α, β; λz2 × 2 Supersin2 α+β α+β+1 , ; 4 2 , 2

(3.380)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions 

z

t α−1 (z − t)β−1 cos (λt (z − t)) dt = B (α, β) zα+β−1   α, β; λz2 × 2 Supercos 2 α+β α+β+1 , ; 4 2 , 2  z t α−1 (z − t)β−1 sinh (λt (z − t)) dt = B (α, β) zα+β−1 0   α, β; λz2 × 2 Supersinh2 α+β α+β+1 , ; 4 2 , 2 0

and

197



(3.381)

(3.382)

z

t α−1 (z − t)β−1 cosh (λt (z − t)) dt = B (α, β) zα+β−1   α, β; λz2 × 2 Supercosh2 α+β α+β+1 , ; 4 2 , 2 0

(3.383)

where λ is a constant. Proof. Using the integral representations  z t α−1 (z − t)β−1 eiλt(z−t) dt = B (α, β) zα+β−1 0   α, β; iλz2 × 2 F2 α+β α+β+1 , ; 4 2 , 2  z t α−1 (z − t)β−1 e−iλt(z−t) dt = B (α, β) zα+β−1 0   α, β; −iλz2 × 2 F2 α+β α+β+1 , ; 4 2 , 2  z t α−1 (z − t)β−1 eλt(z−t) dt = B (α, β) zα+β−1 0   α, β; λz2 × 2 F2 α+β α+β+1 , ; 4 2 , 2 and



(3.384)

(3.385)

(3.386)

z

t α−1 (z − t)β−1 e−λt(z−t) dt = B (α, β) zα+β−1   α, β; λz2 × 2 F2 α+β α+β+1 − , 4 ; 2 , 2 0

where Re (α) > 0, Re (β) > 0, and |z| < 1, we obtain the results.

(3.387)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 3.55. If Re (α) > 0, Re (β) > 0, and |z| < 1, then we have (see Chapter 2) 

  t α−1 (z − t)β−1 sin λt (z − t)2 dt = B (α, β) zα+β−1 0   α, β2 , β+1 4 2 2 ; × 3 Supersin3 α+β α+β+1 α+β+2 λz , , 3 ;9 3 , 3  z   t α−1 (z − t)β−1 cos λt (z − t)2 dt = B (α, β) zα+β−1 0   α, β2 , β+1 4 2 2 ; × 3 Supercos 3 α+β α+β+1 α+β+2 λz , , 3 ;9 3 , 3  z   t α−1 (z − t)β−1 sinh λt (z − t)2 dt = B (α, β) zα+β−1 0   α, β2 , β+1 4 2 2 ; × 3 Supersinh3 α+β α+β+1 α+β+2 λz , , , ;9 z

3

3

(3.388)

(3.389)

(3.390)

3

and 

  t α−1 (z − t)β−1 cosh λt (z − t)2 dt = B (α, β) zα+β−1 0   α, β2 , β+1 4 2 2 ; × 3 Supercosh3 α+β α+β+1 α+β+2 λz , , 3 ;9 3 , 3 z

(3.391)

where λ is a constant. Proof. By means of the equalities 

z

2

t α−1 (z − t)β−1 eiλt(z−t) dt = B (α, β) zα+β−1 0   α, β2 , β+1 4 2 2 ; × 3 F3 α+β α+β+1 α+β+2 iλz , , 3 ;9 3 , 3  z 2 t α−1 (z − t)β−1 e−iλt(z−t) dt = B (α, β) zα+β−1 0   α, β2 , β+1 4 2 2 ; × 3 F3 α+β α+β+1 α+β+2 − iλz , 9 , 3 ; 3 , 3  z 2 t α−1 (z − t)β−1 eλt(z−t) dt = B (α, β) zα+β−1 0   α, β2 , β+1 4 2 2 ; × 3 F3 α+β α+β+1 α+β+2 λz , , , ;9 3

3

3

(3.392)

(3.393)

(3.394)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions 

z

t α−1 (z − t)β−1 e−λt(z−t) dt = B (α, β) zα+β−1 0   α, β2 , β+1 ; 4 × 3 F3 α+β α+β+1 2 α+β+2 − λz2 , 9 , 3 ; 3 , 3

199

2

where Re (α) > 0, Re (β) > 0, and |z| < 1, we obtain the results. If Re (α) > 0, Re (β) > 0, and |z| < 1, then we have (see Chapter 2)    z β α, α+1 4 2 2 , 2; α−1 β−1 λt 2 (z−t) α+β−1 λz , t e dt = B (α, β) z (z − t) 3 F3 α+β α+β+1 α+β+2 9 , , ; 0 3 3 3

(3.395)

(3.396)

where λ is a constant. Theorem 3.56. If Re (α) > 0, Re (β) > 0, and |z| < 1, then  z   t α−1 (z − t)β−1 sin λt 2 (z − t) dt = B (α, β) zα+β−1 0   β α, α+1 4 2 2 , 2; × 3 Supersin3 α+β α+β+1 α+β+2 λz , , 3 ;9 3 , 3  z   t α−1 (z − t)β−1 cos λt 2 (z − t) dt = B (α, β) zα+β−1 0   β α, α+1 4 2 2 , 2; × 3 Supercos 3 α+β α+β+1 α+β+2 λz , , 3 ;9 3 , 3  z   t α−1 (z − t)β−1 sinh λt 2 (z − t) dt = B (α, β) zα+β−1 0   β α, α+1 4 2 2 , 2; × 3 Supersinh3 α+β α+β+1 α+β+2 λz , , , ;9 3

and



3

(3.397)

(3.398)

(3.399)

3

  t α−1 (z − t)β−1 cosh λt 2 (z − t) dt = B (α, β) zα+β−1 0   β α, α+1 4 2 2 , 2; × 3 Supercosh3 α+β α+β+1 α+β+2 λz , , 3 ;9 3 , 3 z

(3.400)

where λ is a constant. Theorem 3.57. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then  z t α−1 (z − t)β−1 1 Supersin0 (a1 ; −; λt (z − t))dt = B (α, β) zα+β−1 0   a1 , α, β; 1 2 × 3 Supersin2 α+β α+β+1 λz , ;4 2 , 2

(3.401)

200

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

z

t α−1 (z − t)β−1 1 Supercos 0 (a1 ; −; λt (z − t))dt = B (α, β) zα+β−1   a1 , α, β; 1 2 × 3 Supercos 2 α+β α+β+1 λz , ;4 2 , 2  z t α−1 (z − t)β−1 1 Supersinh0 (a1 ; −; λt (z − t))dt = B (α, β) zα+β−1 0   a1 , α, β; 1 2 × 3 Supersinh2 α+β α+β+1 λz , ;4 2 , 2 0

and



(3.402)

(3.403)

z

t α−1 (z − t)β−1 1 Supercosh0 (a1 ; −; λt (z − t))dt = B (α, β) zα+β−1   a1 , α, β; 1 2 × 3 Supercosh2 α+β α+β+1 λz , ;4 2 , 2 0

(3.404)

where λ is a constant. Proof. Using the equalities  z t α−1 (z − t)β−1 1 F0 (a1 ; −; iλt (z − t))dt = B (α, β) zα+β−1 0   a1 , α, β; 1 2 × 3 F2 α+β α+β+1 iλz , ;4 2 , 2  z t α−1 (z − t)β−1 1 F0 (a1 ; −; −iλt (z − t))dt = B (α, β) zα+β−1 0   a1 , α, β; 1 2 × 3 F2 α+β α+β+1 − iλz , 4 ; 2 , 2  z t α−1 (z − t)β−1 1 F0 (a1 ; −; λt (z − t))dt = B (α, β) zα+β−1 0   a1 , α, β; 1 2 × 3 F2 α+β α+β+1 λz , ;4 2 , 2 and



(3.405)

(3.406)

(3.407)

z

t α−1 (z − t)β−1 1 F0 (a1 ; −; −λt (z − t))dt = B (α, β) zα+β−1   a1 , α, β; 1 2 × 3 F2 α+β α+β+1 − λz , 4 ; 2 , 2 0

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, we deduce the results.

(3.408)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

201

Theorem 3.58. If Re (α) > 0, Re (β) > 0, and |z| < 1, then 

z

t α−1 (z − t)β−1 1 Supersin0 (a1 ; −; λ (z − t))dt = B (α, β) zα+β−1 0   a1 , β; × 2 Supersin1 λz , α + β;  z t α−1 (z − t)β−1 1 Supercos 0 (a1 ; −; λ (z − t))dt = B (α, β) zα+β−1 0   a1 , β; λz , × 2 Supercos 1 α + β;  z t α−1 (z − t)β−1 1 Supersinh0 (a1 ; −; λ (z − t))dt = B (α, β) zα+β−1 0   a1 , β; × 2 Supersinh1 λz , α + β;

(3.409)

(3.410)

(3.411)

and 

z

t α−1 (z − t)β−1 1 Supercosh0 (a1 ; −; λ (z − t))dt = B (α, β) zα+β−1   a1 , β; λz , × 2 Supercosh1 α + β; 0

(3.412)

where λ is a constant. Proof. Using the equalities 

z

t α−1 (z − t)β−1 1 F0 (a1 ; −; iλ (z − t))dt   a1 , β; α+β−1 iλz , = B (α, β) × z 2 F1 α + β;  z t α−1 (z − t)β−1 1 F0 (a1 ; −; −iλ (z − t))dt 0   a1 , β; α+β−1 = B (α, β) × z − iλz , 2 F1 α + β;  z t α−1 (z − t)β−1 1 F0 (a1 ; −; λ (z − t))dt 0   a1 , β; α+β−1 λz , = B (α, β) × z 2 F1 α + β; 0

(3.413)

(3.414)

(3.415)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and 

z

t α−1 (z − t)β−1 1 F0 (a1 ; −; −λ (z − t))dt   a1 , β; − λz , = B (α, β) × zα+β−1 2 F1 α + β; 0

(3.416)

where Re (α) > 0, Re (β) > 0, and |z| < 1, we obtain the results. Theorem 3.59. If Re (α) > 0, Re (β) > 0, and |z| < 1, then 

  t α−1 (z − t)β−1 sin λt 2 dt = zα+β−1 0  α α+1  2, 2 ; 2 × 2 Supersin2 α+β α+β+1 λz , ; 2 , 2  z   1 t α−1 (z − t)β−1 cos λt 2 dt = zα+β−1 B (α, β) 0  α α+1  2, 2 ; 2 × 2 Supercos 2 α+β α+β+1 λz , ; 2 , 2  z   1 t α−1 (z − t)β−1 sinh λt 2 dt = zα+β−1 B (α, β) 0  α α+1  2, 2 ; 2 × 2 Supersinh2 α+β α+β+1 λz , ; 2 , 2 1 B (α, β)

z

(3.417)

(3.418)

(3.419)

and 

  t α−1 (z − t)β−1 cosh λt 2 dt = zα+β−1 0  α α+1  2, 2 ; 2 × 2 Supercosh2 α+β α+β+1 λz , ; 2 , 2

1 B (α, β)

z

(3.420)

where λ is a constant. Proof. In view of the integral expressions 1 B (α, β)



z

2

t α−1 (z − t)β−1 eiλt dt 0  α α+1  2, 2 ; α+β−1 2 =z × 2 F2 α+β α+β+1 iλz , ; 2 , 2

(3.421)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

1 B (α, β)



z

t α−1 (z − t)β−1 e−iλt dt 0  α α+1  , ; 2 2 = zα+β−1 × 2 F2 α+β α+β+1 − iλz2 , ; 2 , 2  z 1 2 t α−1 (z − t)β−1 eλt dt B (α, β) 0  α α+1  , ; 2 2 = zα+β−1 × 2 F2 α+β α+β+1 λz2 , ; 2 , 2

203

2

(3.422)

(3.423)

and 1 B (α, β)



z

t α−1 (z − t)β−1 e−λt dt 0  α α+1  2, 2 ; α+β−1 2 =z × 2 F2 α+β α+β+1 − λz , ; 2 , 2 2

(3.424)

where Re (α) > 0, Re (β) > 0, and |z| < 1, we have the results.

3.6 Analytic number theory via Gauss hypergeometric functions In this section, we consider the analytic number theory involving the Gauss hypergeometric functions.

3.6.1 Analytic number theory for new special functions We now consider the analytic number theory for the new special functions. Theorem 3.60. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then 

  t α−1 (1 − t)β−1 1 Supersin0 a1 ; −; λt 2 dt 0   a1 , α2 , α+1 2 ; = 3 Supersin2 α+β α+β+1 λ , ; 2 , 2  1   1 t α−1 (1 − t)β−1 1 Supercos 0 a1 ; −; λt 2 dt B (α, β) 0   a1 , α2 , α+1 2 ; = 3 Supercos 2 α+β α+β+1 λ , ; 2 , 2 1 B (α, β)

1

(3.425)

(3.426)

204

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

  t α−1 (1 − t)β−1 1 Supersinh0 a1 ; −; λt 2 dt 0   a1 , α2 , α+1 2 ; = 3 Supersinh2 α+β α+β+1 λ , ; 2 , 2

1 B (α, β)

1

(3.427)

and 

  t α−1 (1 − t)β−1 1 Supercosh0 a1 ; −; λt 2 dt 0   a1 , α2 , α+1 2 ; = 3 Supercosh2 α+β α+β+1 λ , ; 2 , 2

1 B (α, β)

1

(3.428)

where λ is a constant. Proof. By means of the relations 

  t α−1 (1 − t)β−1 1 F0 a1 ; −; iλt 2 dt 0  a1 , α2 , α+1 2 ; = 3 F2 α+β α+β+1 iλ , ; 2 , 2  1   1 t α−1 (1 − t)β−1 1 F0 a1 ; −; −iλt 2 dt B (α, β) 0   ; a1 , α2 , α+1 2 = 3 F2 α+β α+β+1 − iλ , , ; 2 2  1   1 t α−1 (1 − t)β−1 1 F0 a1 ; −; λt 2 dt B (α, β) 0   a1 , α2 , α+1 2 ; = 3 F2 α+β α+β+1 λ , ; 2 , 2 1 B (α, β) 

1

(3.429)

(3.430)

(3.431)

and 1 B (α, β)  = 3 F2



  t α−1 (1 − t)β−1 1 F0 a1 ; −; −λt 2 dt 0  a1 , α2 , α+1 2 ; α+β α+β+1 − λ , ; 2 , 2 1

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, we deduce the results.

(3.432)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

205

Theorem 3.61. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and |z| < 1, then 

1

t α−1 (1 − t)β−1 2 Supersin1 (a1 , a2 ; c1 ; λ (1 − t))dt = B (α, β) 0   a1 , a2 , β; λ , × 3 Supersin2 c1 , α + β;  1 t α−1 (1 − t)β−1 2 Supercos 1 (a1 , a2 ; c1 ; λ (1 − t))dt = B (α, β) 0   a1 , a2 , β; × 3 Supercos 2 λ , c1 , α + β;  1 t α−1 (1 − t)β−1 2 Supersinh1 (a1 , a2 ; c1 ; λ (1 − t))dt = B (α, β) 0   a1 , a2 , β; × 3 Supersinh2 λ , c1 , α + β;

(3.433)

(3.434)

(3.435)

and 

1

t α−1 (1 − t)β−1 2 Supercosh1 (a1 , a2 ; c1 ; λ (1 − t))dt = B (α, β) 0   a1 , a2 , β; × 3 Supercosh2 λ , c1 , α + β;

(3.436)

where λ is a constant. Proof. By means of the integral representations 

1

t α−1 (1 − t)β−1 2 F1 (a1 , a2 ; c1 ; λ (1 − t))dt   a1 , a2 , β; λ , = B (α, β) × 3 F2 c1 , α + β;  1 t α−1 (1 − t)β−1 2 F1 (a1 , a2 ; c1 ; λ (1 − t))dt 0   a1 , a2 , β; λ , = B (α, β) × 3 F2 c1 , α + β;  1 t α−1 (1 − t)β−1 2 F1 (a1 , a2 ; c1 ; λ (1 − t))dt 0   a1 , a2 , β; λ , = B (α, β) × 3 F2 c1 , α + β; 0

(3.437)

(3.438)

(3.439)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and 

1

t α−1 (1 − t)β−1 2 F1 (a1 , a2 ; c1 ; λ (1 − t))dt   a1 , a2 , β; λ , = B (α, β) × 3 F2 c1 , α + β; 0

(3.440)

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and |z| < 1, we deduce the results. Theorem 3.62. If Re (α) > 0, Re (β) > 0, and |z| < 1, then 

  t α−1 (1 − t)β−1 sin λ (1 − t)2 dt = B (α, β) 0   β β+1 2, 2 ; × 2 Supersin2 α+β α+β+1 λ , ; 2 , 2  1   t α−1 (1 − t)β−1 cos λ (1 − t)2 dt = B (α, β) 0   β β+1 2, 2 ; × 2 Supercos 2 α+β α+β+1 λ , ; 2 , 2  1   t α−1 (1 − t)β−1 sinh λ (1 − t)2 dt = B (α, β) 0   β β+1 , ; 2 × 2 Supersinh2 α+β2 α+β+1 λ , , ; 2 2 1

(3.441)

(3.442)

(3.443)

and 

  t α−1 (1 − t)β−1 cosh λ (1 − t)2 dt = B (α, β) 0   β β+1 2, 2 ; × 2 Supercosh2 α+β α+β+1 λ , ; 2 , 2 1

(3.444)

where λ is a constant. Proof. In view of 

1

2

t α−1 (1 − t)β−1 eiλ(1−t) dt 0   β β+1 2, 2 ; = B (α, β) × 2 F2 α+β α+β+1 iλ , ; 2 , 2

(3.445)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions 

1

t α−1 (1 − t)β−1 e−iλ(1−t) dt 0   β β+1 2, 2 ; = B (α, β) × 2 F2 α+β α+β+1 − iλ , ; 2 , 2  1 2 t α−1 (1 − t)β−1 eλ(1−t) dt 0   β β+1 , ; 2 = B (α, β) × 2 F2 α+β2 α+β+1 λ , , ; 2 2

207

2

(3.446)

(3.447)

and 

1

t α−1 (1 − t)β−1 e−λ(1−t) dt 0   β β+1 2, 2 ; = B (α, β) × 2 F2 α+β α+β+1 − λ , ; 2 , 2 2

(3.448)

where Re (α) > 0, Re (β) > 0, and |z| < 1, we have the results. Theorem 3.63. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, s ∈ N, and |z| < 1, then 

1

 t α−1 (1 − t)β−1 1 Supersin2 a1 ; c1 , c2 ; λ (1 − t)s dt

0

= B (α, β)



× 1+s Supersin2+s 

1

β+s−1 a1 , βs , β+1 ; s ,··· , s λ α+β α+β+1 α+β+s−1 c1 , c2 , s , s , · · · , ; s

(3.449)

 ,

 t α−1 (1 − t)β−1 1 Supercos 2 a1 ; c1 , c2 ; λ (1 − t)s dt

0

= B (α, β)



× 1+s Supercos 2+s 

1

β+s−1 a1 , βs , β+1 ; s ,··· , s

α+β+1 , · · · , α+β+s−1 ; c1 , c2 , α+β s , s s

(3.450)

 λ ,

 t α−1 (1 − t)β−1 1 Supersinh2 a1 ; c1 , c2 ; λ (1 − t)s dt

0

= B (α, β) × 1+s Supersinh2+s



β+s−1 a1 , βs , β+1 ; s ,··· , s λ α+β α+β+1 α+β+s−1 ; c1 , c2 , s , s , · · · , s

(3.451)

 ,

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and 

1

 t α−1 (1 − t)β−1 1 Supercosh2 a1 ; c1 , c2 ; λ (1 − t)s dt

0

= B (α, β)



× 1+s Supercosh2+s

β+s−1 a1 , βs , β+1 ; s ,··· , s λ α+β α+β+1 α+β+s−1 c1 , c2 , s , s , · · · , ; s

(3.452)

 ,

where λ is a constant. Proof. Using the integral representations 

1

 t α−1 (1 − t)β−1 1 F2 a1 ; c1 , c2 ; iλ (1 − t)s dt

0

= B (α, β)



β+s−1 a1 , βs , β+1 ; s ,··· , s iλ α+β α+β+1 α+β+s−1 c1 , c2 , s , s , · · · , ; s

× 1+s F2+s 

1

(3.453)

 ,

 t α−1 (1 − t)β−1 1 F2 a1 ; c1 , c2 ; −iλ (1 − t)s dt

0

= B (α, β)



× 1+s F2+s 

1

β+s−1 a1 , βs , β+1 ; s ,··· , s α+β α+β+1 α+β+s−1 c1 , c2 , s , s , · · · , ; s



(3.454)

− iλ ,

 t α−1 (1 − t)β−1 1 F2 a1 ; c1 , c2 ; λ (1 − t)s dt

0

= B (α, β)



× 1+s F2+s



β+s−1 a1 , βs , β+1 ; s ,··· , s

α+β+1 , · · · , α+β+s−1 ; c1 , c2 , α+β s , s s

(3.455)

λ ,

and 

1

 t α−1 (1 − t)β−1 1 F2 a1 ; c1 , c2 ; −λ (1 − t)s dt

0

= B (α, β) × 1+s F2+s



β+s−1 a1 , βs , β+1 ; s ,··· , s α+β α+β+1 α+β+s−1 ; c1 , c2 , s , s , · · · , s



(3.456)

−λ ,

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, s ∈ N, and |z| < 1, we deduce the results.

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

209

Theorem 3.64. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, s ∈ N, and |z| < 1, then there are  1

 t α−1 (1 − t)β−1 2 Supersin1 a1 , a2 ; c1 ; λ (1 − t)s dt = B (α, β) 0   (3.457) β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s × 2+s Supersin1+s λ , α+β+1 , · · · , α+β+s−1 ; c1 , α+β s , s s  1

 t α−1 (1 − t)β−1 2 Supercos 1 a1 , a2 ; c1 ; λ (1 − t)s dt = B (α, β) 0   (3.458) β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s × 2+s Supercos 1+s λ , α+β+1 , · · · , α+β+s−1 ; c1 , α+β s , s s  1

 t α−1 (1 − t)β−1 2 Supersinh1 a1 , a2 ; c1 ; λ (1 − t)s dt 0

= B (α, β)



× 2+s Supersinh1+s and



1

β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s λ α+β α+β+1 α+β+s−1 c1 , s , s , · · · , ; s

(3.459)

 ,

 t α−1 (1 − t)β−1 2 Supercosh1 a1 , a2 ; c1 ; λ (1 − t)s dt

0

= B (α, β)



× 2+s Supercosh1+s



β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s α+β+1 c1 , α+β , · · · , α+β+s−1 ; s , s s

(3.460)

λ ,

where λ is a constant. Proof. Using the integral representations  1

 t α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; iλ (1 − t)s dt 0

= B (α, β)



× 2+s F1+s 

1

β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s iλ α+β α+β+1 α+β+s−1 c1 , s , s , · · · , ; s

(3.461)

 ,

 t α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; −iλ (1 − t)s dt

0

= B (α, β) × 2+s F1+s



β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s α+β+1 , · · · , α+β+s−1 ; c1 , α+β s , s s

 − iλ ,

(3.462)

210

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

1

 t α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; λ (1 − t)s dt

0

= B (α, β)



× 2+s F1+s



β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s α+β+1 c1 , α+β , · · · , α+β+s−1 ; s , s s

(3.463)

λ ,

and 

1

 t α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; −λ (1 − t)s dt

0

= B (α, β)



× 2+s F1+s

β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s α+β α+β+1 α+β+s−1 ; c1 , s , s , · · · , s



(3.464)

−λ ,

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, s ∈ N, and |z| < 1, we obtain the result. Theorem 3.65. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, then 1 B (α, β)



1

t α−1 (1 − t)β−1 1 Supersin0 (a1 ; −; λt)dt 0   a1 , α; λ , = 2 Supersin1 α + β;  z 1 t α−1 (z − t)β−1 1 Supercos 0 (a1 ; −; λt)dt B (α, β) 0   a1 , α; = 2 Supercos 1 λz , α + β;  1 1 t α−1 (1 − t)β−1 1 Supersinh0 (a1 ; −; λt)dt B (α, β) 0   a1 , α; = 2 Supersinh1 λ , α + β;

(3.465)

(3.466)

(3.467)

and 1 B (α, β)



1

t α−1 (1 − t)β−1 1 Supercosh0 (a1 ; −; λt)dt   a1 , α; λ , = 2 Supercosh1 α + β;

where λ is a constant.

0

(3.468)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

Proof. In view of the integral relations    1 1 a1 , α; t α−1 (1 − t)β−1 1 F0 (a1 ; −; iλt)dt = 2 F1 iλ , α + β; B (α, β) 0    1 1 a1 , α; − iλ , t α−1 (1 − t)β−1 1 F0 (a1 ; −; −iλt)dt = 2 F1 α + β; B (α, β) 0    1 1 a1 , α; α−1 β−1 t λ , (1 − t) 1 F0 (a1 ; −; λt)dt = 2 F1 α + β; B (α, β) 0 and 1 B (α, β)



1

211

(3.469) (3.470) (3.471)

 t α−1 (1 − t)β−1 1 F0 (a1 ; −; −λt)dt = 2 F1

0

 a1 , α; −λ , α + β;

(3.472)

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, and |z| < 1, we obtain the results. Theorem 3.66. If Re (α) > 0, Re (β) > 0, and |z| < 1, then 

 λ t sin (λt (1 − t)) dt = B (α, β)2 Supersin2 α+β α+β+1 , (1 − t) ;4 0 2 , 2    1 α, β; λ α−1 β−1 t cos (λt (1 − t)) dt = B (α, β)2 Supercos 2 α+β α+β+1 , (1 − t) ;4 0 2 , 2    1 α, β; λ t α−1 (1 − t)β−1 sinh (λt (1 − t)) dt = B (α, β)2 Supersinh2 α+β α+β+1 , ;4 0 2 , 2 

and 

1

α−1

α, β;

β−1



1

t

α−1

(1 − t)

β−1

0

cosh (λt (1 − t)) dt = B (α, β)2 Supercosh2

 λ α+β α+β+1 4 , ; 2 , 2 α, β;

(3.473)

(3.474)

(3.475)

(3.476)

where λ is a constant. Proof. Use the integral representations  1 t α−1 (1 − t)β−1 eiλt(z−t) dt = B (α, β) zα+β−1 0   α, β; λ × 2 F2 α+β α+β+1 i , ;4 2 , 2  1 t α−1 (1 − t)β−1 e−iλt(z−t) dt = B (α, β) zα+β−1 0   α, β; λ × 2 F2 α+β α+β+1 − i , 4 ; 2 , 2

(3.477)

(3.478)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

1

t α−1 (1 − t)β−1 eλt(z−t) dt = B (α, β) zα+β−1 0   α, β; λ × 2 F2 α+β α+β+1 , ;4 2 , 2

(3.479)

and 

1

t α−1 (1 − t)β−1 e−λt(z−t) dt = B (α, β) zα+β−1   α, β; λ × 2 F2 α+β α+β+1 − , 4 ; 2 , 2 0

(3.480)

where λ is a constant.

Theorem 3.67. If Re (α) > 0 and Re (β) > 0, then we have 

  t α−1 (1 − t)β−1 sin λt (1 − t)2 dt = B (α, β) 0   α, β2 , β+1 4 2 ; × 3 Supersin3 α+β α+β+1 α+β+2 λ , , 3 ;9 3 , 3  1   t α−1 (1 − t)β−1 cos λt (1 − t)2 dt = B (α, β) 0   α, β2 , β+1 ; 4 × 3 Supercos 3 α+β α+β+1 2 α+β+2 λ , , 3 ;9 3 , 3  1   t α−1 (1 − t)β−1 sinh λt (1 − t)2 dt = B (α, β) 0   α, β2 , β+1 4 2 ; × 3 Supersinh3 α+β α+β+1 α+β+2 λ , , , ;9 1

3

3

(3.481)

(3.482)

(3.483)

3

and 

  t α−1 (1 − t)β−1 cosh λt (1 − t)2 dt = B (α, β) 0   α, β2 , β+1 4 2 ; × 3 Supercosh3 α+β α+β+1 α+β+2 λ , , 3 ;9 3 , 3

where λ is a constant.

1

(3.484)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

213

Proof. With the integral expressions 

1

2

t α−1 (1 − t)β−1 eiλt(1−t) dt = B (α, β) 0   α, β2 , β+1 4 2 ; × 3 F3 α+β α+β+1 α+β+2 λi , , 3 ;9 3 , 3  1 2 t α−1 (1 − t)β−1 e−iλt(1−t) dt = B (α, β) 0   α, β2 , β+1 4 2 ; × 3 F3 α+β α+β+1 α+β+2 − λi , 9 , 3 ; 3 , 3  1 2 t α−1 (1 − t)β−1 eλt(1−t) dt = B (α, β) 0   α, β2 , β+1 4 2 ; × 3 F3 α+β α+β+1 α+β+2 λ , , , ;9 3

3

(3.485)

(3.486)

(3.487)

3

and 

1

t α−1 (1 − t)β−1 e−λt(1−t) dt = B (α, β) 0   α, β2 , β+1 4 2 ; × 3 F3 α+β α+β+1 α+β+2 − λ , 9 , 3 ; 3 , 3 2

(3.488)

where Re (α) > 0 and Re (β) > 0, we deduce the results. Theorem 3.68. If Re (α) > 0 and Re (β) > 0, then 

  t α−1 (1 − t)β−1 sin λt 2 (1 − t) dt = B (α, β) 0   β α, α+1 4 2 , 2; × 3 Supersin3 α+β α+β+1 α+β+2 λ , , 3 ;9 3 , 3  1   t α−1 (1 − t)β−1 cos λt 2 (1 − t) dt = B (α, β) 0   β α, α+1 4 2 , 2; × 3 Supercos 3 α+β α+β+1 α+β+2 λ , , 3 ;9 3 , 3  1   t α−1 (1 − t)β−1 sinh λt 2 (1 − t) dt = B (α, β) 0   β α, α+1 4 2 , 2; × 3 Supersinh3 α+β α+β+1 α+β+2 λ , , , ;9 1

3

3

3

(3.489)

(3.490)

(3.491)

214

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and 

  t α−1 (1 − t)β−1 cosh λt 2 (1 − t) dt = B (α, β) 0   β α, α+1 4 2 , 2; × 3 Supercosh3 α+β α+β+1 α+β+2 λ , , 3 ;9 3 , 3 1

(3.492)

where λ is a constant. Proof. With the relations 

1

0

t α−1 (1 − t)β−1 eiλt 

= B (α, β) × 3 F3 

1

0

 0

1

β α, α+1 2 , 2;

2 (1−t)

β α, α+1 2 , 2;

2 (1−t)

(3.493)

dt

 4 − λi , α+β α+β+1 α+β+2 9 , 3 ; 3 , 3

t α−1 (1 − t)β−1 eλt 

= B (α, β) × 3 F3

dt

 4 λi , α+β α+β+1 α+β+2 9 , 3 ; 3 , 3

t α−1 (1 − t)β−1 e−iλt 

= B (α, β) × 3 F3

2 (1−t)

(3.494)

dt

 4 λ , α+β α+β+1 α+β+2 9 , 3 ; 3 , 3 β α, α+1 2 , 2;

(3.495)

and  0

1

t α−1 (1 − t)β−1 e−λt 

= B (α, β) × 3 F3

2 (1−t)

dt

 4 − λ , α+β α+β+1 α+β+2 9 , 3 ; 3 , 3 β α, α+1 2 , 2;

(3.496)

where Re (α) > 0 and Re (β) > 0, we obtain the results. Theorem 3.69. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then 

1

t α−1 (1 − t)β−1 1 Supersin0 (a1 ; −; λt (1 − t))dt = B (α, β)   a1 , α, β; 1 × 3 Supersin2 α+β α+β+1 λ , ;4 2 , 2 0

(3.497)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions 

1

t α−1 (1 − t)β−1 1 Supercos 0 (a1 ; −; λt (1 − t))dt = B (α, β) 0   a1 , α, β; 1 × 3 Supercos 2 α+β α+β+1 λ , ;4 2 , 2  1 t α−1 (1 − t)β−1 1 Supersinh0 (a1 ; −; λt (1 − t))dt = B (α, β) 0   a1 , α, β; 1 × 3 Supersinh2 α+β α+β+1 λ , ;4 2 , 2 and



215

(3.498)

(3.499)

1

t α−1 (1 − t)β−1 1 Supercosh0 (a1 ; −; λt (1 − t))dt = B (α, β)   a1 , α, β; 1 × 3 Supercosh2 α+β α+β+1 λ , ;4 2 , 2 0

(3.500)

where λ is a constant. Proof. By means of the integral expressions  1 t α−1 (1 − t)β−1 1 F0 (a1 ; −; iλt (1 − t))dt = B (α, β) 0   a1 , α, β; 1 × 3 F2 α+β α+β+1 λi , ;4 2 , 2  1 t α−1 (1 − t)β−1 1 F0 (a1 ; −; −iλt (1 − t))dt = B (α, β) 0   a1 , α, β; 1 × 3 F2 α+β α+β+1 − λi , 4 ; 2 , 2  1 t α−1 (1 − t)β−1 1 F0 (a1 ; −; λt (1 − t))dt = B (α, β) 0   a1 , α, β; 1 × 3 F2 α+β α+β+1 λ , ;4 2 , 2 and



(3.501)

(3.502)

(3.503)

1

t α−1 (1 − t)β−1 1 F0 (a1 ; −; −λt (1 − t))dt = B (α, β)   a1 , α, β; 1 × 3 F2 α+β α+β+1 − λ , 4 ; 2 , 2 0

where Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, we have the results.

(3.504)

216

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 3.70. If Re (α) > 0 and Re (β) > 0, then 

1

t α−1 (1 − t)β−1 1 Supersin0 (a1 ; −; λ (1 − t))dt = B (α, β) 0   a1 , β; λ , × 2 Supersin1 α + β;  1 t α−1 (1 − t)β−1 1 Supercos 0 (a1 ; −; λ (1 − t))dt = B (α, β) 0   a1 , β; × 2 Supercos 1 λ , α + β;  1 t α−1 (1 − t)β−1 1 Supersinh0 (a1 ; −; λ (1 − t))dt = B (α, β) 0   a1 , β; × 2 Supersinh1 λ , α + β;

(3.505)

(3.506)

(3.507)

and 

1

t α−1 (1 − t)β−1 1 Supercosh0 (a1 ; −; λ (1 − t))dt = B (α, β) 0   a1 , β; × 2 Supercosh1 λ , α + β;

(3.508)

where λ is a constant. Proof. With the equalities 

1

t α−1 (1 − t)β−1 1 F0 (a1 ; −; iλ (1 − t))dt = B (α, β)   a1 , β; × 2 F1 λi , α + β;  1 t α−1 (1 − t)β−1 1 F0 (a1 ; −; −iλ (1 − t))dt = B (α, β) 0   a1 , β; × 2 F1 − iλ , α + β;  1 t α−1 (1 − t)β−1 1 F0 (a1 ; −; λ (1 − t))dt = B (α, β) 0   a1 , β; × 2 F1 λ , α + β; 0

(3.509)

(3.510)

(3.511)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

217

and 

1

t α−1 (1 − t)β−1 1 F0 (a1 ; −; −λ (1 − t))dt = B (α, β)   a1 , β; × 2 F1 −λ , α + β; 0

(3.512)

where Re (α) > 0 and Re (β) > 0, we obtain the results. Theorem 3.71. If Re (α) > 0 and Re (β) > 0, then 

1 B (α, β)

α−1

(1 − t)

α−1

(1 − t)

t

β−1



2

sin λt



 dt = 2 Supersin2

0



1 B (α, β) 1 B (α, β)

1

1

t

β−1

 cos λt

2





dt = 2 Supercos 2

0



1

α−1

(1 − t)

α−1

(1 − t)

t

  sinh λt 2 dt = 2 Supersinh2

β−1

α α+1 2, 2 ; α+β α+β+1 λ ; 2 , 2



α α+1 2, 2 ; α+β α+β+1 λ ; 2 , 2



0

,

(3.513)

,

(3.514)

 

α α+1 2, 2 ; α+β α+β+1 λ ; 2 , 2

,

(3.515)

,

(3.516)

and 1 B (α, β)



1

t

β−1



2

cosh λt



 dt = 2 Supercosh2

0

α α+1 2, 2 ; α+β α+β+1 λ ; 2 , 2



where λ is a constant. Proof. Using the integral representations 

1 B (α, β) 1 B (α, β)



1

t

α−1

β−1 iλt 2

(1 − t)

e

dt = 2 F2

0



1

t

 α−1

(1 − t)

β−1 −iλt 2

e

dt = 2 F2

0

1 B (α, β)



t

α−1

β−1 λt 2

(1 − t)

e

dt = 2 F2

0

and 1 B (α, β)





1

t

α α+1 2, 2 ; α+β α+β+1 ; 2 , 2



1

α−1

(1 − t)

β−1 −λt 2

e

dt = 2 F2

0

where Re (α) > 0 and Re (β) > 0, we deduce the results.

α α+1 2, 2 ; α+β α+β+1 iλ ; 2 , 2

,

(3.517)



− iλ ,

α α+1 2, 2 ; λ α+β α+β+1 ; 2 , 2

α α+1 2, 2 ; α+β α+β+1 ; 2 , 2



(3.518)

 ,

(3.519)

 −λ ,

(3.520)

218

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 3.72. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then 

  (t + 1)α−1 (1 − t)β−1 1 Supersin0 a1 ; −; λ (t + 1)2 dt −1   a1 , α2 , α+1 2 ; α+β−1 =2 × 3 Supersin2 α+β α+β+1 4λ , ; 2 , 2  1   1 (t + 1)α−1 (1 − t)β−1 1 Supercos 0 a1 ; −; λ (t + 1)2 dt B (α, β) −1   α α+1 , , ; a 1 2 2 = 2α+β−1 × 3 Supercos 2 α+β α+β+1 4λ , ; 2 , 2  1   1 (t + 1)α−1 (1 − t)β−1 1 Supersinh0 a1 ; −; λ (t + 1)2 dt B (α, β) −1   a1 , α2 , α+1 2 ; α+β−1 =2 × 3 Supersinh2 α+β α+β+1 4λ , ; 2 , 2 1 B (α, β)

1

(3.521)

(3.522)

(3.523)

and 

  (t + 1)α−1 (1 − t)β−1 1 Supercosh0 a1 ; −; λ (t + 1)2 dt −1   a1 , α2 , α+1 2 ; α+β−1 =2 × 3 Supercosh2 α+β α+β+1 4λ , ; 2 , 2

1 B (α, β)

1

(3.524)

where λ is a constant. Proof. In view of the integral relations 

  (t + 1)α−1 (1 − t)β−1 1 F0 a1 ; −; iλ (t + 1)2 dt −1   a1 , α2 , α+1 2 ; α+β−1 =2 × 3 F2 α+β α+β+1 4λi , ; 2 , 2  1   1 (t + 1)α−1 (1 − t)β−1 1 F0 a1 ; −; −iλ (t + 1)2 dt B (α, β) −1   a1 , α2 , α+1 2 ; α+β−1 =2 × 3 F2 α+β α+β+1 − 4λi , ; 2 , 2  1   1 (t + 1)α−1 (1 − t)β−1 1 F0 a1 ; −; λ (t + 1)2 dt B (α, β) −1   a1 , α2 , α+1 2 ; α+β−1 =2 × 3 F2 α+β α+β+1 4λ , ; 2 , 2 1 B (α, β)

1

(3.525)

(3.526)

(3.527)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

219

and 

  (t + 1)α−1 (1 − t)β−1 1 F0 a1 ; −; −λ (t + 1)2 dt −1   a1 , α2 , α+1 2 ; α+β−1 =2 × 3 F2 α+β α+β+1 − 4λ , ; 2 , 2

1 B (α, β)

1

(3.528)

where Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, we obtain the results. Theorem 3.73. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, and Re (c1 ) > 0, then 

1

(t + 1)α−1 (1 − t)β−1 2 Supersin1 (a1 , a2 ; c1 ; λ (1 − t))dt

−1



= B (α, β) 2 

1



= B (α, β) 2 1 −1

× 3 Supersin2

 a1 , a2 , β; 2λ , c1 , α + β;

(3.529)

(t + 1)α−1 (1 − t)β−1 2 Supercos 1 (a1 , a2 ; c1 ; λ (1 − t))dt

−1



α+β−1

α+β−1

× 3 Supercos 2

 a1 , a2 , β; 2λ , c1 , α + β;

(3.530)

(t + 1)α−1 (1 − t)β−1 2 Supersinh1 (a1 , a2 ; c1 ; λ (1 − t))dt 

= B (α, β) 2

α+β−1

× 3 Supersinh2

 a1 , a2 , β; 2λ , c1 , α + β;

(3.531)

and 

1

−1

(t + 1)α−1 (1 − t)β−1 2 Supercosh1 (a1 , a2 ; c1 ; λ (1 − t))dt 

= B (α, β) z

α+β−1

× 3 Supercosh2

 a1 , a2 , β; 2λ , c1 , α + β;

(3.532)

where λ is a constant. Proof. In view of 

1

−1

(t + 1)α−1 (1 − t)β−1 2 F1 (a1 , a2 ; c1 ; iλ (1 − t))dt

= B (α, β) 2

 α+β−1

× 3 F2

 a1 , a2 , β; 2λi , c1 , α + β;

(3.533)

220

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

1

(t + 1)α−1 (1 − t)β−1 2 F1 (a1 , a2 ; c1 ; −iλ (1 − t))dt

−1



= B (α, β) 2 

1

−1

α+β−1

× 3 F2

 a1 , a2 , β; − 2λi , c1 , α + β;

(t + 1)α−1 (1 − t)β−1 2 F1 (a1 , a2 ; c1 ; λ (1 − t))dt 

 a1 , a2 , β; 2λ , c1 , α + β;

= B (α, β) 2α+β−1 × 3 F2 and



1

−1

(3.534)

(3.535)

(t + 1)α−1 (1 − t)β−1 2 F1 (a1 , a2 ; c1 ; −λ (1 − t))dt

= B (α, β) 2

 α+β−1

× 3 F2

 a1 , a2 , β; − 2λ , c1 , α + β;

(3.536)

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, and Re (c1 ) > 0, we have the results. Theorem 3.74. If Re (α) > 0, Re (β) > 0, and |z| < 1, then  1   (t + 1)α−1 (1 − t)β−1 sin λ (1 − t)2 dt = B (α, β) 2α+β−1 −1   β β+1 2, 2 ; × 2 Supersin2 α+β α+β+1 4λ , ; 2 , 2  1   (t + 1)α−1 (1 − t)β−1 cos λ (1 − t)2 dt = B (α, β) 2α+β−1 −1   β β+1 2, 2 ; × 2 Supercos 2 α+β α+β+1 4λ , ; 2 , 2  1   (t + 1)α−1 (1 − t)β−1 sinh λ (1 − t)2 dt = B (α, β) 2α+β−1 −1   β β+1 2, 2 ; × 2 Supersinh2 α+β α+β+1 4λ , ; 2 , 2 and



  (t + 1)α−1 (1 − t)β−1 cosh λ (1 − t)2 dt = B (α, β) 2α+β−1 −1   β β+1 2, 2 ; × 2 Supercosh2 α+β α+β+1 4λ , ; 2 , 2

(3.537)

(3.538)

(3.539)

1

where λ is a constant.

(3.540)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

221

Proof. Due to the integral representations 



1

2

(t + 1)α−1 (1 − t)β−1 eiλ(1−t) dt −1   β β+1 2, 2 ; α+β−1 = B (α, β) 2 × 2 F2 α+β α+β+1 4λi , ; 2 , 2 1

(3.541)

(t + 1)α−1 (1 − t)β−1 e−iλ(1−t) dt −1   β β+1 , ; 2 = B (α, β) 2α+β−1 × 2 F2 α+β2 α+β+1 − 4λi , , ; 2 2  1 2 (t + 1)α−1 (1 − t)β−1 eλ(1−t) dt −1   β β+1 2, 2 ; α+β−1 = B (α, β) 2 × 2 F2 α+β α+β+1 4λ , ; 2 , 2 2

(3.542)

(3.543)

and 

1

(t + 1)α−1 (1 − t)β−1 e−λ(1−t) dt −1   β β+1 2, 2 ; α+β−1 = B (α, β) 2 × 2 F2 α+β α+β+1 − 4λ , ; 2 , 2 2

(3.544)

where Re (α) > 0 and Re (β) > 0, we deduce the results. Theorem 3.75. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, and s ∈ N, then 

1 −1

 (t + 1)α−1 (1 − t)β−1 1 Supersin2 a1 ; c1 , c2 ; λ (1 − t)s dt

= B (α, β) 2α+β−1 × 1+s Supersin2+s 

1

−1



β+s−1 a1 , βs , β+1 ; s ,··· , s α+β+1 c1 , c2 , α+β ,··· s , s

,

α+β+s−1 ; s



(3.545)

λ2s ,

 (t + 1)α−1 (1 − t)β−1 1 Supercos 2 a1 ; c1 , c2 ; λ (1 − t)s dt

= B (α, β) 2α+β−1 × 1+s Supercos 2+s



β+s−1 a1 , βs , β+1 ; s ,··· , s α+β+1 ,··· c1 , c2 , α+β s , s

,

α+β+s−1 ; s

 λ2s ,

(3.546)

222

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

1

−1

 (t + 1)α−1 (1 − t)β−1 1 Supersinh2 a1 ; c1 , c2 ; λ (1 − t)s dt

= B (α, β) 2α+β−1 × 1+s Supersinh2+s and



1

−1



β+s−1 a1 , βs , β+1 ; s ,··· , s λ2s α+β α+β+1 α+β+s−1 ; c1 , c2 , s , s , · · · , s

(3.547)

 ,

 (t + 1)α−1 (1 − t)β−1 1 Supercosh2 a1 ; c1 , c2 ; λ (1 − t)s dt

= B (α, β) 2α+β−1 × 1+s Supercosh2+s



β+s−1 a1 , βs , β+1 ; s ,··· , s λ2s α+β α+β+1 α+β+s−1 ; c1 , c2 , s , s , · · · , s

(3.548)

 ,

where λ is a constant. Proof. By means of



1

−1



 (t + 1)α−1 (1 − t)β−1 1 F2 a1 ; c1 , c2 ; iλ (1 − t)s dt

= B (α, β) 2α+β−1   β+s−1 a1 , βs , β+1 , · · · , ; s s × 1+s F2+s λ2s i , α+β+1 α+β+s−1 , , · · · , ; c1 , c2 , α+β s s s 1

−1

(3.549)

 (t + 1)α−1 (1 − t)β−1 1 F2 a1 ; c1 , c2 ; −iλ (1 − t)s dt

= B (α, β) 2α+β−1   β+s−1 a1 , βs , β+1 ; s ,··· , s s − λ2 i , × 1+s F2+s α+β+1 c1 , c2 , α+β , · · · , α+β+s−1 ; s , s s  1

 (t + 1)α−1 (1 − t)β−1 1 F2 a1 ; c1 , c2 ; λ (1 − t)s dt

(3.550)

= B (α, β) 2α+β−1   β+s−1 a1 , βs , β+1 ; s ,··· , s s × 1+s F2+s λ2 , α+β+1 c1 , c2 , α+β , · · · , α+β+s−1 ; s , s s

(3.551)

−1

and



1 −1

 (t + 1)α−1 (1 − t)β−1 1 F2 a1 ; c1 , c2 ; −λ (1 − t)s dt

= B (α, β) 2α+β−1   β+s−1 a1 , βs , β+1 ; s ,··· , s s − λ2 , × 1+s F2+s α+β+1 , · · · , α+β+s−1 ; c1 , c2 , α+β s , s s

(3.552)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

223

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (c1 ) > 0, Re (c2 ) > 0, and s ∈ N, we obtain the results. Theorem 3.76. If Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and s ∈ N, then 

1

−1

 (t + 1)α−1 (1 − t)β−1 2 Supersin1 a1 , a2 ; c1 ; λ (1 − t)s dt

= B (α, β) 2α+β−1



× 2+s Supersin1+s 

1

−1



× 2+s Supercos 1+s 1

−1

(3.553)

 ,

 (t + 1)α−1 (1 − t)β−1 2 Supercos 1 a1 , a2 ; c1 ; λ (1 − t)s dt

= B (α, β) 2α+β−1



β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s λ2s α+β α+β+1 α+β+s−1 ; c1 , s , s , · · · , s

β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s λ2s α+β α+β+1 α+β+s−1 c1 , s , s , · · · , ; s

(3.554)

 ,

 (t + 1)α−1 (1 − t)β−1 2 Supersinh1 a1 , a2 ; c1 ; λ (1 − t)s dt

= B (α, β) 2α+β−1 × 2+s Supersinh1+s



β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s λ2s α+β α+β+1 α+β+s−1 ; c1 , s , s , · · · , s

(3.555)

 ,

and 

1

−1

 (t + 1)α−1 (1 − t)β−1 2 Supercosh1 a1 , a2 ; c1 ; λ (1 − t)s dt

= B (α, β) 2α+β−1 × 2+s Supercosh1+s



β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s λ2s α+β α+β+1 α+β+s−1 c1 , s , s , · · · , ; s

(3.556)

 ,

where λ is a constant. Proof. By means of the integral relations 

1

−1

 (t + 1)α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; iλ (1 − t)s dt

= B (α, β) 2α+β−1   β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s s λ2 i , × 2+s F1+s α+β+1 , · · · , α+β+s−1 ; c1 , α+β s , s s

(3.557)

224

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

 (t + 1)α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; −iλ (1 − t)s dt

1 −1

= B (α, β) 2α+β−1   β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s s × 2+s F1+s − λ2 i , α+β+1 , · · · , α+β+s−1 ; c1 , α+β s , s s  1

 (t + 1)α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; λ (1 − t)s dt

(3.558)

= B (α, β) 2α+β−1   β+s−1 a1 , a2 , βs , β+1 ; s ,··· , s s × 2+s F1+s λ2 , α+β+1 α+β+s−1 , , · · · , ; c1 , α+β s s s

(3.559)

−1

and



1

−1

 (t + 1)α−1 (1 − t)β−1 2 F1 a1 , a2 ; c1 ; −λ (1 − t)s dt

= B (α, β) 2α+β−1   β+s−1 a1 , a2 , βs , β+1 , · · · , ; s s × 2+s F1+s − λ2s , α+β+1 α+β+s−1 , , · · · , ; c1 , α+β s s s

(3.560)

where Re (α) > 0, Re (β) > 0, Re (a1 ) > 0, Re (a2 ) > 0, Re (c1 ) > 0, and s ∈ N, we have the results. Theorem 3.77. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then  1 1 (t + 1)α−1 (1 − t)β−1 1 Supersin0 (a1 ; −; λ (t + 1))dt B (α, β) −1 = 2α+β−1



× 2 Supersin1 1 B (α, β)



1 −1

a1 , α; 2λ , α + β;



× 2 Supercos 1 

1 −1

(3.561)

(t + 1)α−1 (1 − t)β−1 1 Supercos 0 (a1 ; −; λ (t + 1))dt

= 2α+β−1

1 B (α, β)





(3.562)

a1 , α; 2λ , α + β;

(t + 1)α−1 (1 − t)β−1 1 Supersinh0 (a1 ; −; λ (t + 1))dt

= 2α+β−1 × 2 Supersinh1





a1 , α; 2λ , α + β;

(3.563)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

225

and 

1 B (α, β)

1

−1

(t + 1)α−1 (1 − t)β−1 1 Supercosh0 (a1 ; −; λ (t + 1))dt

= 2α+β−1



× 2 Supercosh1

 a1 , α; 2λ , α + β;

(3.564)

where λ is a constant. Proof. Using the integral relations 

1 B (α, β) =2

−1



1 B (α, β) =2

1

−1

α+β−1

(t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; iλ (t + 1))dt 

× 2 F1

α+β−1

1 B (α, β) =2

1



1

−1

α+β−1

(3.565)

(t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; −λ (t + 1) i)dt  a1 , α; − 2λi , α + β;

× 2 F1 

 a1 , α; 2λi , α + β;

(3.566)

(t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; λ (t + 1))dt 

× 2 F1

 a1 , α; 2λ , α + β;

(3.567)

and 1 B (α, β)



1

−1

(t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; −λ (t + 1))dt

= 2α+β−1 × 2 F1



 a1 , α; − 2λ , α + β;

(3.568)

where Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, we obtain the results. Theorem 3.78. If Re (α) > 0 and Re (β) > 0, then 

1

(t + 1)α−1 (1 − t)β−1 sin (λ (1 + t) (1 − t)) dt = B (α, β) 2α+β−1   α, β; × 2 Supersin2 α+β α+β+1 λ , ; 2 , 2 −1

(3.569)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

1

(t + 1)α−1 (1 − t)β−1 cos (λ (1 + t) (1 − t)) dt = B (α, β) 2α+β−1 −1   α, β; × 2 Supercos 2 α+β α+β+1 λ , ; 2 , 2  1 (t + 1)α−1 (1 − t)β−1 sinh (λ (1 + t) (1 − t)) dt = B (α, β) 2α+β−1 −1   α, β; × 2 Supersinh2 α+β α+β+1 λ , ; 2 , 2 and



(3.570)

(3.571)

1

(t + 1)α−1 (1 − t)β−1 cosh (λ (1 + t) (1 − t)) dt = B (α, β) 2α+β−1   α, β; × 2 Supercosh2 α+β α+β+1 λ , ; 2 , 2 −1

(3.572)

where λ is a constant. Proof. Applying the integral expressions  1 (t + 1)α−1 (1 − t)β−1 eiλ(1+t)(1−t) dt = B (α, β) 2α+β−1 −1   α, β; × 2 F2 α+β α+β+1 λi , ; 2 , 2  1 (t + 1)α−1 (1 − t)β−1 e−iλ(1+t)(1−t) dt = B (α, β) 2α+β−1 −1   α, β; × 2 F2 α+β α+β+1 − iλ , ; 2 , 2  1 (t + 1)α−1 (1 − t)β−1 eλ(1+t)(1−t) dt = B (α, β) 2α+β−1 −1   α, β; × 2 F2 α+β α+β+1 λ , ; 2 , 2 and



(3.573)

(3.574)

(3.575)

1

(t + 1)α−1 (1 − t)β−1 e−λ(1+t)(1−t) dt = B (α, β) 2α+β−1   α, β; × 2 F2 α+β α+β+1 − λ , ; 2 , 2 −1

where Re (α) > 0 and Re (β) > 0, we deduce the results.

(3.576)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

227

Theorem 3.79. If Re (α) > 0 and Re (β) > 0, then we have (see Chapter 2) 

  (t + 1)α−1 (1 − t)β−1 sin λ (1 + t) (1 − t)2 dt = B (α, β) 2α+β−1 −1   α, β2 , β+1 16 2 ; λ , × 3 Supersin3 α+β α+β+1 α+β+2 , 3 ; 9 3 , 3  1   (t + 1)α−1 (1 − t)β−1 cos λ (1 + t) (1 − t)2 dt = B (α, β) 2α+β−1 −1   α, β2 , β+1 16 2 ; × 3 Supercos 3 α+β α+β+1 α+β+2 λ , , 3 ; 9 3 , 3  1   (t + 1)α−1 (1 − t)β−1 sinh λ (1 + t) (1 − t)2 dt = B (α, β) 2α+β−1 −1   α, β2 , β+1 ; 16 λ , × 3 Supersinh3 α+β α+β+1 2 α+β+2 , , ; 9 1

3

3

(3.577)

(3.578)

(3.579)

3

and 

  (t + 1)α−1 (1 − t)β−1 cosh λ (1 + t) (1 − t)2 dt = B (α, β) 2α+β−1 −1   α, β2 , β+1 16 2 ; λ , × 3 Supercosh3 α+β α+β+1 α+β+2 , , ; 9 1

3

3

(3.580)

3

where λ is a constant. Proof. For Re (α) > 0 and Re (β) > 0, we have the integral representations 

1

−1

2

(t + 1)α−1 (1 − t)β−1 eiλ(1+t)(1−t) dt

= B (α, β) 2α+β−1   α, β2 , β+1 16 2 ; λi , × 3 F3 α+β α+β+1 α+β+2 , 3 ; 9 3 , 3  1 2 (t + 1)α−1 (1 − t)β−1 e−iλ(1+t)(1−t) dt

(3.581)

= B (α, β) 2α+β−1   α, β2 , β+1 16 2 ; × 3 F3 α+β α+β+1 α+β+2 − λi , 9 , 3 ; 3 , 3

(3.582)

−1

228

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

1

2

−1

(t + 1)α−1 (1 − t)β−1 eλ(1+t)(1−t) dt

= B (α, β) 2α+β−1   α, β2 , β+1 16 2 ; λ , × 3 F3 α+β α+β+1 α+β+2 , 3 ; 9 3 , 3

(3.583)

and 

1

−1

(t + 1)α−1 (1 − t)β−1 e−λ(1+t)(1−t) dt 2

= B (α, β) 2α+β−1   α, β2 , β+1 16 2 ; × 3 F3 α+β α+β+1 α+β+2 − λ , 9 , 3 ; 3 , 3

(3.584)

from which we conclude the results. Theorem 3.80. If Re (α) > 0 and Re (β) > 0, then 

  (t + 1)α−1 (1 − t)β−1 sin λ (1 + t)2 (1 − t) dt = B (α, β) 2α+β−1 −1   β α, α+1 16 2 , 2; λ , × 3 Supersin3 α+β α+β+1 α+β+2 9 , , ; 3 3 3  1   (t + 1)α−1 (1 − t)β−1 cos λ (1 + t)2 (1 − t) dt = B (α, β) 2α+β−1 −1   β α, α+1 16 2 , 2; λ , × 3 Supercos 3 α+β α+β+1 α+β+2 , 3 ; 9 3 , 3  1   (t + 1)α−1 (1 − t)β−1 sinh λ (1 + t)2 (1 − t) dt = B (α, β) 2α+β−1 −1   β α, α+1 16 2 , 2; λ , × 3 Supersinh3 α+β α+β+1 α+β+2 , , ; 9 1

3

3

(3.585)

(3.586)

(3.587)

3

and 

  (t + 1)α−1 (1 − t)β−1 cosh λ (1 + t)2 (1 − t) dt = B (α, β) 2α+β−1 −1   β α, α+1 16 2 , 2; λ , × 3 Supercosh3 α+β α+β+1 α+β+2 , 3 ; 9 3 , 3 1

where λ is a constant.

(3.588)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

229

Proof. For Re (α) > 0 and Re (β) > 0, we find the integral representations 

1

2

(t + 1)α−1 (1 − t)β−1 eiλ(1+t) (1−t) dt = B (α, β) 2α+β−1 −1   β α, α+1 16 2 , 2; × 3 F3 α+β α+β+1 α+β+2 λi , , 3 ; 9 3 , 3  1 2 (t + 1)α−1 (1 − t)β−1 e−iλ(1+t) (1−t) dt = B (α, β) 2α+β−1 −1   β , ; α, α+1 16 2 2 − λi , × 3 F3 α+β α+β+1 α+β+2 9 , , ; 3 3 3  1 2 (t + 1)α−1 (1 − t)β−1 eλ(1+t) (1−t) dt = B (α, β) 2α+β−1 −1   β α, α+1 16 2 , 2; λ , × 3 F3 α+β α+β+1 α+β+2 , , ; 9 3

3

(3.589)

(3.590)

(3.591)

3

and 

1

(t + 1)α−1 (1 − t)β−1 e−λ(1+t) (1−t) dt = B (α, β) 2α+β−1 −1   β α, α+1 16 2 , 2; × 3 F3 α+β α+β+1 α+β+2 − λ , 9 , 3 ; 3 , 3 2

(3.592)

which yield the results. Theorem 3.81. If Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, then 

1

(t + 1)α−1 (1 − t)β−1 1 Supersin0 (a1 ; −; λ (1 + t) (1 − t))dt   a1 , α, β; α+β−1 = B (α, β) 2 × 3 Supersin2 α+β α+β+1 λ , ; 2 , 2  1 (t + 1)α−1 (1 − t)β−1 1 Supercos 0 (a1 ; −; λ (1 + t) (1 − t))dt −1   a1 , α, β; α+β−1 = B (α, β) 2 × 3 Supercos 2 α+β α+β+1 λ , ; 2 , 2  1 (t + 1)α−1 (1 − t)β−1 1 Supersinh0 (a1 ; −; λ (1 + t) (1 − t))dt −1   a1 , α, β; α+β−1 = B (α, β) 2 × 3 Supersinh2 α+β α+β+1 λ , ; 2 , 2 −1

(3.593)

(3.594)

(3.595)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and 

1

(t + 1)α−1 (1 − t)β−1 1 Supercosh0 (a1 ; −; λ (1 + t) (1 − t))dt   , α, β; a 1 = B (α, β) 2α+β−1 × 3 Supercosh2 α+β α+β+1 λ , ; 2 , 2 −1

(3.596)

where λ is a constant. Proof. With the help of the integral relations 

1

(t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; iλ (1 + t) (1 − t))dt = B (α, β) 2α+β−1 −1   a1 , α, β; × 3 F2 α+β α+β+1 λi , ; 2 , 2  1 (t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; −iλ (1 + t) (1 − t))dt = B (α, β) 2α+β−1 −1   a1 , α, β; × 3 F2 α+β α+β+1 − iλ , ; 2 , 2  1 (t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; λ (1 + t) (1 − t))dt = B (α, β) 2α+β−1 −1   a1 , α, β; × 3 F2 α+β α+β+1 λ , ; 2 , 2

(3.597)

(3.598)

(3.599)

and 

1

(t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; −λ (1 + t) (1 − t))dt = B (α, β) 2α+β−1   a1 , α, β; × 3 F2 α+β α+β+1 − λ , ; 2 , 2 −1

(3.600)

where Re (α) > 0, Re (β) > 0, and Re (a1 ) > 0, we deduce the results. Theorem 3.82. If Re (α) > 0 and Re (β) > 0, then 

1

−1

(t + 1)α−1 (1 − t)β−1 1 Supersin0 (a1 ; −; λ (1 − t))dt

= B (α, β) 2α+β−1   a1 , β; 2λ , × 2 Supersin1 α + β;

(3.601)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions 

1

−1

231

(t + 1)α−1 (1 − t)β−1 1 Supercos 0 (a1 ; −; λ (1 − t))dt

= B (α, β) 2α+β−1   a1 , β; × 2 Supercos 1 2λ , α + β;  1 (t + 1)α−1 (1 − t)β−1 1 Supersinh0 (a1 ; −; λ (1 − t))dt

(3.602)

= B (α, β) 2α+β−1   a1 , β; 2λ , × 2 Supersinh1 α + β;

(3.603)

−1

and



1 −1

(t + 1)α−1 (1 − t)β−1 1 Supercosh0 (a1 ; −; λ (1 − t))dt

= B (α, β) 2α+β−1   a1 , β; × 2 Supercosh1 2λ , α + β;

(3.604)

where λ is a constant. Proof. Using the relations  1

(t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; iλ (1 − t))dt   a1 , β; = B (α, β) 2α+β−1 × 2 F1 2λi , α + β;  1 (t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; −iλ (1 − t))dt −1   a1 , β; = B (α, β) 2α+β−1 × 2 F1 − 2λi , α + β;  1 (t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; λ (1 − t))dt −1   a1 , β; α+β−1 2λ , = B (α, β) 2 × 2 F1 α + β; −1

and



(3.605)

(3.606)

(3.607)

1

(t + 1)α−1 (1 − t)β−1 1 F0 (a1 ; −; −λ (1 − t))dt −1   a1 , β; = B (α, β) 2α+β−1 × 2 F1 − 2λ , α + β; where Re (α) > 0 and Re (β) > 0, we obtain the results.

(3.608)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 3.83. If Re (α) > 0 and Re (β) > 0, then 

  (t + 1)α−1 (1 − t)β−1 sin λ (t + 1)2 dt = 2α+β−1 −1   α α+1 2, 2 ; × 2 Supersin2 α+β α+β+1 4λ , ; 2 , 2  1   1 (t + 1)α−1 (1 − t)β−1 cos λ (t + 1)2 dt = 2α+β−1 B (α, β) −1   α α+1 2, 2 ; × 2 Supercos 2 α+β α+β+1 4λ , ; 2 , 2  1   1 (t + 1)α−1 (1 − t)β−1 sinh λ (t + 1)2 dt = 2α+β−1 B (α, β) −1   α α+1 2, 2 ; × 2 Supersinh2 α+β α+β+1 4λ , ; 2 , 2 1 B (α, β)

1

(3.609)

(3.610)

(3.611)

and 

  (t + 1)α−1 (1 − t)β−1 cosh λ (t + 1)2 dt = 2α+β−1 −1   α α+1 2, 2 ; × 2 Supercosh2 α+β α+β+1 4λ , ; 2 , 2

1 B (α, β)

1

(3.612)

where λ is a constant. Proof. In view of the integral representations 1 B (α, β)



1

2

(t + 1)α−1 (1 − t)β−1 eiλ(t+1) dt −1   α α+1 2, 2 ; α+β−1 =2 × 2 F2 α+β α+β+1 4λi , ; 2 , 2  1 1 2 (t + 1)α−1 (1 − t)β−1 e−iλ(t+1) dt B (α, β) −1   α α+1 2, 2 ; α+β−1 =2 × 2 F2 α+β α+β+1 − 4λi , ; 2 , 2  1 1 2 (t + 1)α−1 (1 − t)β−1 eλ(t+1) dt B (α, β) −1   α α+1 2, 2 ; α+β−1 =2 × 2 F2 α+β α+β+1 4λ , ; 2 , 2

(3.613)

(3.614)

(3.615)

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

233

and 1 B (α, β)



1

(t + 1)α−1 (1 − t)β−1 e−λ(t+1) dt −1   α α+1 2, 2 ; α+β−1 =2 × 2 F2 α+β α+β+1 − 4λ , ; 2 , 2 2

(3.616)

where Re (α) > 0 and Re (β) > 0, we deduce the results. For more detail of the Gauss hypergeometric series, see Rainville (1960) [45], Gasper (2004) [69], Fine (1988) [68], Bailey (1935) [60], Ono (1998) [111], Ahlgreen and Ono (2000) [112], Ahlgren (2001) [113], Osburn and Schneider (2009) [114], Greene and Stanton (1986) [115], Mortenson (2005) [116], Barman and Kalita (2012) [117], Mortenson (2003) [118], Frechette et al. (2004) [119], Lennon (2011) [120], Sadek and El-Sissi (2016) [121], Wang et al. (2016) [122], Ancarani and Gasaneo (2009) [123], and so on.

3.6.2 The results for the zeros via new special functions Without proofs, using the definitions of the new special functions, we present the following results on their zeros. Theorem 3.84. 2 Supersin1 (1, 1; 2; 0)



1 = lim (Log (1 − iz) − Log (1 + iz)) z→0 2z

(3.617)

= 0. Theorem 3.85. 2 Supercos 1 (1, 1; 2; 0)

 1 = lim − (Log (1 − iz) + Log (1 + z)) z→0 2zi

(3.618)

= 1. Theorem 3.86. 2 Supertan1 (1, 1; 2; 0) = 0.

(3.619)

2 Supersec 1 (1, 1; 2; 0) = 1.

(3.620)

Theorem 3.87.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 3.88. 2 Supersinh1 (1, 1; 2; 0)



1 = lim (Log (1 + z) − Log (1 − z)) z→0 2z

(3.621)

= 0. Theorem 3.89. 2 Supercosh1 (1, 1; 2; 0)



1 = lim − (Log (1 + z) + Log (1 − z)) z→0 2z

(3.622)

= 1. Theorem 3.90. 2 Supertanh1 (1, 1; 2; 0) = 0.

(3.623)

2 Supersech1 (1, 1; 2; 0) = 1.

(3.624)

1 Supersin0 (a; −; 0) = 0,

(3.625)

1 Supercos 0 (a; −; 0) = 1,

(3.626)

1 Supertan0 (a; −; 0) = 0,

(3.627)

1 Supersec 0 (a; −; 0) = 1,

(3.628)

1 Supersinh0 (a; −; 0) = 0,

(3.629)

Theorem 3.91.

Theorem 3.92.

where Re (a) > 0. Theorem 3.93.

where Re (a) > 0. Theorem 3.94.

where Re (a) > 0. Theorem 3.95.

where Re (a) > 0. Theorem 3.96.

where Re (a) > 0.

Chapter 3 • Hypergeometric supertrigonometric and superhyperbolic functions

235

Theorem 3.97. 1 Supercosh0 (a; −; 0) = 1,

(3.630)

1 Supertanh0 (a; −; 0) = 0,

(3.631)

1 Supersech0 (a; −; 0) = 1,

(3.632)

0 Supersin1 (−; c; 0) = 0,

(3.633)

0 Supercos 1 (−; c; 0) = 1,

(3.634)

0 Supertan1 (−; c; 0) = 0,

(3.635)

0 Supersec 1 (−; c; 0) = 1,

(3.636)

0 Supersinh1 (−; c; 0) = 0,

(3.637)

0 Supercosh1 (−; c; 0) = 1,

(3.638)

where Re (a) > 0. Theorem 3.98.

where Re (a) > 0. Theorem 3.99.

where Re (a) > 0. Theorem 3.100.

where Re (c) > 0. Theorem 3.101.

where Re (c) > 0. Theorem 3.102.

where Re (c) > 0. Theorem 3.103.

where Re (c) > 0. Theorem 3.104.

where Re (c) > 0. Theorem 3.105.

where Re (c) > 0.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 3.106. 0 Supertanh1 (−; c; 0) = 0,

(3.639)

0 Supersech1 (−; c; 0) = 1,

(3.640)

where Re (c) > 0. Theorem 3.107.

where Re (c) > 0.

4 Hypergeometric supertrigonometric and superhyperbolic functions via Kummer confluent hypergeometric series 4.1 The Kummer confluent hypergeometric series of first type In this section, we introduce the definition, properties and theorems for the Kummer confluent hypergeometric series of first type and present the Laplace and Mellin transforms and integral representations for the Kummer confluent hypergeometric series of first type.

4.1.1 Definition, properties and theorems for the Kummer confluent hypergeometric series of first type We start with the definition for the Kummer confluent hypergeometric series of first type. Definition 4.1. The Kummer confluent hypergeometric series of first type is defined as [124] 1 F1 (a; c; z)



= 1 F1

a ;z c



a (a + 1) z2 a + ··· =1+ z+ c c (c + 1) 2 ∞  (a)n zn = , (c)n n!

(4.1)

n=0

where a, c ∈ C, n ∈ N, and |z| < 1. The series (4.1) was introduced by Kummer in 1836 [124]. It converges locally uniformly in C to an entire function [51]. An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions https://doi.org/10.1016/B978-0-12-824154-7.00010-6 Copyright © 2021 Elsevier Inc. All rights reserved.

237

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Property 4.1 (The confluent hypergeometric differential equation). The hypergeometric series (4.1) is a solution of the confluent hypergeometric differential equation (see [83], p. 292) z

d 2ϕ dϕ + (b − z) − aϕ = 0, dz dz2

(4.2)

where a, z ∈ C, c ∈ C\Z− 0 , and |z| < 1. The result was presented in [83, p. 292]. Property 4.2. The hypergeometric series of the form (see [125], p. 504) 1 F1 (a; c; λz)

=1+ =

∞  n=0

a a (a + 1) (λz)2 + ··· (λz) + c c (c + 1) 2

(4.3)

n

(a)n (λz) , (c)n n!

is a solution of the differential equation z

dϕ d 2ϕ − aλϕ = 0, + (b − λz) 2 dz dz

(4.4)

where a, λ, z ∈ C, c ∈ C\Z− 0 , and |z| < 1. The property is presented by Abramowitz and Stegun in 1972 [125]. For a, c, λ ∈ C and n ∈ N, we have 1 F1 (a; c; λz)

=1+ =

a a (a + 1) (λz)2 + ··· (λz) + c c (c + 1) 2

∞  (a)n (λz)n n=0

n!

(c)n

(4.5)

,

which leads to the series representations 1 F1 (a; c; −iλz)

=1+ =

a a (a + 1) (−iλz)2 + ··· (−iλz) + c c (c + 1) 2

∞  (a)n (−iλz)n n=0

(c)n

n!

,

(4.6)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

239

1 F1 (a; c; iλz)

=1+ =

a a (a + 1) (iλz)2 + ··· (iλz) + c c (c + 1) 2

∞  (a)n (iλz)n n=0

(c)n

n!

(4.7)

,

and 1 F1 (a; c; −λz)

=1+ =

∞  (a)n (−λz)n n=0

where i =

a a (a + 1) (−λz)2 + ··· (−λz) + c c (c + 1) 2

(c)n

n!

(4.8)

,

√ −1.

Theorem 4.1. [Abramowitz and Stegun (1972)] We have  1  (c) F t a−1 (1 − t)c−a−1 ezt dt c; z) = (a; 1 1  (a)  (c − a) 0 and 1 F1 (a; c; λz) =

 (c)  (a)  (c − a)



1

t a−1 (1 − t)c−a−1 eλzt dt,

(4.9)

(4.10)

0

where λ ∈ C and Re (c) > Re (a). The result was presented by Abramowitz and Stegun in 1972 [125]. A particular case was reported by Rainville in 1960, developed by Manocha and Srivastava in 1984 [22], and introduced by Andrews, Askey, and Roy in 1999 [25]. Theorem 4.2. [Rainville (1960)] If Re (c) > Re (a) > 0, then we have 1 F1 (a; c; z) =

1  (a)



∞

e−t t a−1 0 F1 (−; c; zt)dt.

(4.11)

0

The result was obtained by Rainville in 1960 (see [45], p. 128). Theorem 4.3. If Re (c) > Re (a) > 0, then we have 

∞

e−t t a−1 0 F1 (−; c; λt)dt =

0

where λ is a constant.

 (c)  (c − a)

 0

1

t a−1 (1 − t)c−a−1 eλt dt,

(4.12)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 4.4. If Re (α) > 0, Re (β) > 0, and |z| < 1, then we have  z 1 t α−1 (z − t)β−1 eλt dt = zα+β−1 B (α, β) 0   α; × 1 F1 λz , α + β;

(4.13)

where λ is a constant. The result was a particular case obtained by Rainville in 1960 [45]. Theorem 4.5. If Re (α) > 0, Re (a) > 0, Re (c) > 0, |z| < 1, Re (c − a) > 0, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 F1 a; c; λt κ dt B (α, β) 0  (4.14) a, ακ , · · · , α+κ−1 ; κ α+β−1 κ =z × 1+κ F1+κ , α+β+κ−1 λz c, α+β ; κ ,··· , κ where λ is a constant. The result is a particular case given by Rainville in 1960 [45]. Proceeding in a similar way, we have the following theorems. Theorem 4.6. If Re (α) > 0, Re (a) > 0, Re (c) > 0, |z| < 1, and Re (c − a) > 0, then we have 

 t α−1 (z − t)β−1 1 F1 a; c; λt 2 dt 0  a, α2 , α+1 2 ; α+β−1 2 =z × 3 F3 , α+β+1 λz c, α+β ; 2 , 2

1 B (α, β)

z

(4.15)

where λ is a constant. Theorem 4.7. If Re (α) > 0, Re (a) > 0, Re (c) > 0, |z| < 1, Re (c − a) > 0, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 F1 a; c; λ (z − t)κ dt = zα+β−1 B (α, β) 0  (4.16) a, βκ , · · · , β+κ−1 ; κ κ λz , × 1+κ F1+κ α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant.

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

241

Theorem 4.8. If Re (α) > 0, Re (a) > 0, Re (c) > 0, |z| < 1, and Re (c − a) > 0, then we have  z

 1 t α−1 (z − t)β−1 1 F1 a; c; λ (z − t)2 dt = zα+β−1 B (α, β) 0  (4.17) a, β2 , β+1 2 ; 2 × 3 F3 λz , α+β+1 c, α+β ; 2 , 2 where λ is a constant. Theorem 4.9. If Re (α) > 0, Re (β) > 0, κ, s ∈ N, Re (c − a) > 0, and |z| < 1, then we have  z   1 t α−1 (z − t)β−1 1 F1 a; c; λt κ (z − t)s dt = zα+β−1 B (α, β) 0  α (4.18) β β+s−1 κ s s λzκ+s , , · · · , ; a, κ , · · · , α+κ−1 κ κ s s × 1+κ+s F1+κ+s , α+β+κ+s−1 (κ + s)κ+s c, α+β ; κ+s , · · · , κ+s where λ is a constant. Theorem 4.10. If Re (α) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 F1 a; c; λt κ dt B (α, β) 0  (4.19) ; a, ακ , · · · , α+κ−1 κ = 1+κ F1+κ λ , α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant. Theorem 4.11. If Re (α) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 F1 a; c; λ (1 − t)κ dt B (α, β) 0  (4.20) a, βκ , · · · , β+κ−1 ; κ = 1+κ F1+κ λ , α+β+κ−1 ; c, α+β κ ,··· , κ where λ is a constant. Theorem 4.12. If Re (α) > 0, Re (β) > 0, κ, s ∈ N, and Re (c − a) > 0, then we have  1   1 t α−1 (1 − t)β−1 1 F1 a; c; λt κ (1 − t)s dt B (α, β) 0  α , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s = 1+κ+s F1+κ+s , α+β+κ+s−1 (κ + s)κ+s c, α+β ; κ+s , · · · , κ+s where λ is a constant. The result is a particular case obtained by Rainville in 1960 [45].

(4.21)

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4.1.2 Laplace and Mellin transforms for the Kummer confluent hypergeometric series of first type In this section, we introduce the Laplace and Mellin transforms for the Kummer confluent hypergeometric series of first type and study the identities via Kummer confluent hypergeometric series of first type. Theorem 4.13 (Laplace transform). Let a, c, z ∈ C and Re (s) > Re (z). Then the Laplace transform of the function t λ−1 1 F1 (a; c; zt) is given as follows:

L t λ−1 1 F1 (a; c; zt)  ∞ e−st 1 F1 (a; c; zt)dt = 0    (λ) λ, a z , = λ 2 F1 ; c s s

(4.22)

where the Laplace transform is defined as 

∞

L [f (t)] = f (s) =

e−st f (t) dt.

(4.23)

0

Theorem 4.14 (Mellin transform). Let a, c, z ∈ C, z = 0, |arg (z)| ≤ π2 , 0 < Re ( ) < Re (a), and 0 < Re ( ) < Re (c). Then the Mellin transform of the function 1 F1 (a; c; −zt) is as follows: M [1 F1 (a; c; −zt)]  ∞ = t −1 1 F1 (a; c; −zt)dt 0

=

(4.24)

 (a − )  (c)  (ω) z− ,  (a)  (c − )

where the Mellin transform is defined as  M [f (t)] = f ( ) =

∞

t −1 f (t) dt.

(4.25)

0

Note that the results are derived from Chapters 2 and 3.

4.2 The hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of first type In this section, based on the Kummer confluent hypergeometric series of first type, we propose the hypergeometric supertrigonometric functions, give the Laplace transforms for

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

243

the hypergeometric supersine and hypergeometric supercosine, and discuss the Laplace transforms for the hypergeometric sine and cosine.

4.2.1 Definitions, properties, and theorems for the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of first type To start with, we give the concepts for the hypergeometric supersine, hypergeometric supercosine, hypergeometric supertangent, hypergeometric supercotangent, hypergeometric supersecant, and hypergeometric supercosecant via Kummer confluent hypergeometric series of first type. Definition 4.2. The hypergeometric supersine via Kummer confluent hypergeometric series of first type is defined as 1 Supersin1 (a; c; z) =

∞  (a)2n+1 (−1)n z2n+1 n=0

(c)2n+1 (2n + 1)!

,

(4.26)

where a, c ∈ C. Definition 4.3. The hypergeometric supercosine via Kummer confluent hypergeometric series of first type is defined as 1 Supercos 1 (a; c; z) =

∞  (a)2n (−1)n z2n n=0

(c)2n

(2n)!

,

(4.27)

where a, c ∈ C. Definition 4.4. The hypergeometric supertangent via Kummer confluent hypergeometric series of first type is defined as 1 Supertan1 (a; c; z) =

1 Supersin1 (a; c; z) 1 Supercos 1 (a; c; z)

,

(4.28)

where a, c ∈ C. Definition 4.5. The hypergeometric supercotangent via Kummer confluent hypergeometric series of first type is defined as 1 Supercot 1 (a; c; z) =

1 Supercos 1 (a; c; z) 1 Supersin1 (a; c; z)

,

(4.29)

where a, c ∈ C. Definition 4.6. The hypergeometric supersecant via Kummer confluent hypergeometric series of first type is defined as 1 Supersec 1 (a; c; z) =

where a, c ∈ C.

1 , Supercos 1 1 (a; c; z)

(4.30)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Definition 4.7. The hypergeometric supercosecant via Kummer confluent hypergeometric series of first type is defined as 1 Supercosec 1 (a; c; z) =

1 , 1 Supersin1 (a; c; z)

(4.31)

where a, c ∈ C. We now present relations among the special functions via Kummer confluent hypergeometric series of first type. By means of the above definitions, we have the hypergeometric supersine via Kummer confluent hypergeometric series of first type 1 Supersin1 (a; c; λz)

1 (1 F1 (a; c; iλz) − 1 F1 (a; c; −iλz)) 2i ∞  (a)2n+1 (−1)n (λz)2n+1 = , (c)2n+1 (2n + 1)!

=

(4.32)

n=0

the hypergeometric supercosine via Kummer confluent hypergeometric series of first type 1 Supercos 1 (a; c; λz)

1 (1 F1 (a; c; iλz) + 1 F1 (a; c; −iλz)) 2 ∞  (a)2n (−1)n (λz)2n , = (c)2n (2n)!

=

(4.33)

n=0

the hypergeometric supertangent via Kummer confluent hypergeometric series of first type 1 Supertan1 (a; c; λz) =

1 Supersin1 (a; c; λz) 1 Supercos 1 (a; c; λz)

,

(4.34)

the hypergeometric supercotangent via Kummer confluent hypergeometric series of first type 1 Supercot 1 (a; c; λz) =

1 Supercos 1 (a; c; λz) 1 Supersin1 (a; c; λz)

,

(4.35)

the hypergeometric supersecant via Kummer confluent hypergeometric series of first type 1 Supersec 1 (a; c; λz) =

1 , 1 Supercos 1 (a; c; λz)

(4.36)

and the hypergeometric supercosecant via Kummer confluent hypergeometric series of first type

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

1 Supercosec 1 (a; c; λz) =

where a, c, λ ∈ C and i =

√ −1.

1 , 1 Supersin1 (a; c; λz)

245

(4.37)

Definition 4.8. The hypergeometric supersine 1 Supersin1 (a; c; z), hypergeometric supercosine 1 Supercos 1 (a; c; z), hypergeometric supertangent 1 Supertan1 (a; c; z), hypergeometric supercotangent 1 Supercot 1 (a; c; z), hypergeometric supersecant 1 Supersec 1 (a; c; z), and hypergeometric supercosecant 1 Supercosec 1 (a; c; z) are called the hypergeometric supertrigonometric functions of the Kummer confluent hypergeometric series of first type. At the moment, it is not difficult to show the following result. √ Theorem 4.15. If a, c, λ ∈ C and i = −1, then we have 1 F1 (a; c; iz) = 1 Supercos 1 (a; c; z) + i 1 Supersin1 (a; c; z)

(4.38)

1 F1 (a; c; −iz) = 1 Supercos 1 (a; c; z) − i 1 Supersin1 (a; c; z) .

(4.39)

1 F1 (a; c; iλz) = 1 Supercos 1 (a; c; λz) + i 1 Supersin1 (a; c; λz)

(4.40)

1 F1 (a; c; iλz) = 1 Supercos 1 (a; c; λz) + i 1 Supersin1 (a; c; λz)

(4.41)

and

Generally,

and

where a, c, λ ∈ C and i =

√

−1.

We now consider the series of the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of first type. Series 1 In this case, we structure series of the form  (a; c; z) =

∞ 

φ (κ) (1 F1 (a; c; iκz)),

(4.42)

κ=0

which can be represented in the form  (a; c; z) = γ +

∞       ϕ (κ) 1 Supercos 1 (a; c; κz) + ψ (κ) 1 Supersin1 (a; c; κz) ,

(4.43)

κ=1

where γ=

ϕ (0) − iψ (0) , 2

(4.44)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

ϕ (κ) − iψ (κ) , 2

φ (κ) =

(4.45)

and ϕ (κ) + iψ (κ) (4.46) 2 are the coefficients of the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of first type. φ (−κ) =

Series 2 If  (a; c; −z) = − (a; c; z), then there exists the series of the form  (a; c; z) =

∞ 

  ψ (κ) 1 Supersin1 (a; c; κz) ,

(4.47)

κ=1

where ψ (κ) are the coefficients of the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of first type. Series 3 If  (a; c; −z) =  (a; c; z), then there exists the series of the form  (a; c; z) = γ +

∞ 

  ϕ (κ) 1 Supercos 1 (a; c; κz) ,

(4.48)

κ=1

where γ and ϕ (κ) are the coefficients of the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of first type.

4.2.2 Integral representations for the Kummer confluent hypergeometric series of first type We now consider the integral representations for the Kummer confluent hypergeometric series of first type. Theorem 4.16. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have 1 Supersin1 (a; c; z) =



 (c)  (a)  (c − a)

where sin (z) =

1

t a−1 (1 − t)c−a−1 sin (z) dt,

(4.49)

0

 1 iz e − e−iz . 2

(4.50)

Proof. In view of the integral relations 1 F1 (a; c; iz) =

 (c)  (a)  (c − a)

 0

1

t a−1 (1 − t)c−a−1 eizt dt

(4.51)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

247

and  (c) 1 F1 (a; c; −iz) =  (a)  (c − a)



1

t a−1 (1 − t)c−a−1 e−izt dt,

(4.52)

0

where Re (c) > Re (a), we have that 1 Supersin1 (a; c; z) ∞  (a)2n+1 (−1)n z2n+1

=

n=0

(c)2n+1 (2n + 1)!

1 (1 F1 (a; c; iz) − 1 F1 (a; c; −iz)) 2 

 1  1 izt  (c) e − e−izt dt = t a−1 (1 − t)c−a−1  (a)  (c − a) 0 2  1  (c) = t a−1 (1 − t)c−a−1 sin (z) dt,  (a)  (c − a) 0 =

(4.53)

where sin (z) =

 1 izt e − e−izt . 2

(4.54)

Theorem 4.17. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have 

 (c) 1 Supercos 1 (a; c; z) =  (a)  (c − a)

1

t a−1 (1 − t)c−a−1 cos (z) dt,

(4.55)

0

where cos (z) =

 1 iz e + e−iz . 2

(4.56)

Proof. With the integral representations 1 F1 (a; c; iz) =

 (c)  (a)  (c − a)



1

t a−1 (1 − t)c−a−1 eizt dt

(4.57)

t a−1 (1 − t)c−a−1 e−izt dt,

(4.58)

0

and  (c) 1 F1 (a; c; −iz) =  (a)  (c − a) where Re (c) > Re (a), we have that

 0

1

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1 Supercos 1 (a; c; z) ∞  (a)2n (−1)n z2n

=

n=0

(c)2n

(2n)!

1 (1 F1 (a; c; iz) + 1 F1 (a; c; −iz)) 2 

 1   (c) a−1 c−a−1 1 izt −izt e +e = dt t (1 − t)  (a)  (c − a) 0 2  1  (c) = t a−1 (1 − t)c−a−1 cos (z) dt,  (a)  (c − a) 0 =

(4.59)

which is derived by the relation cos (z) =

 1 iz e + e−iz . 2

(4.60)

Theorem 4.18. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have  ∞ 1 e−t t a−1 0 Supersin1 (−; c; zt)dt, (4.61) 1 Supersin1 (a; c; z) =  (a) 0 where 0 Supersin1 (−; c; z) =

1 (0 F1 (−; c; iz) − 0 F1 (−; c; −iz)) . 2i

Proof. By means of the integral representations  ∞ 1 F e−t t a−1 0 F1 (−; c; iz)dt c; iz) = (a; 1 1  (a) 0 and 1 F1 (a; c; −iz) =

1  (a)



∞

e−t t a−1 0 F1 (−; c; −iz)dt,

(4.62)

(4.63)

(4.64)

0

we set up the relation 1 Supersin1 (a; c; z) ∞  (a)2n+1 (−1)n z2n+1

=

n=0

(c)2n+1 (2n + 1)!

1 (1 F1 (a; c; iz) − 1 F1 (a; c; −iz)) 2i    ∞ 1 1 = e−t t a−1 (0 F1 (−; c; izt) − 0 F1 (−; c; −izt)) dt  (a) 0 2i  ∞ 1 = e−t t a−1 0 Supersin1 (−; c; zt)dt,  (a) 0 =

(4.65)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

249

where 0 Supersin1 (−; c; z) =

1 (0 F1 (−; c; iz) − 0 F1 (−; c; −iz)) . 2i

(4.66)

Theorem 4.19. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have 1 1 Supercos 1 (a; c; z) =  (a)



∞

e−t t a−1 0 Supercos 1 (−; c; zt)dt,

(4.67)

0

where 0 Supercos 1 (−; c; z) =

1 (0 F1 (−; c; iz) + 0 F1 (−; c; −iz)) . 2

(4.68)

Proof. By means of the integral relations 1 1 F1 (a; c; iz) =  (a) and 1 F1 (a; c; −iz) =

1  (a)



∞

e−t t a−1 0 F1 (−; c; iz)dt

(4.69)

e−t t a−1 0 F1 (−; c; −iz)dt,

(4.70)

0



∞

0

we obtain that 1 Supercos 1 (a; c; z) ∞  (a)2n (−1)n z2n

=

n=0

(c)2n

(2n)!

1 (1 F1 (a; c; iz) + 1 F1 (a; c; −iz)) 2    ∞ 1 −t a−1 1 = e t (0 F1 (−; c; izt) + 0 F1 (−; c; −izt)) dt  (a) 0 2  ∞ 1 = e−t t a−1 0 Supercos 1 (−; c; zt)dt,  (a) 0 =

(4.71)

where 0 Supercos 1 (−; c; z) =

1 (0 F1 (−; c; iz) + 0 F1 (−; c; −iz)) . 2

Without proofs, we suggest the following theorems and properties.

(4.72)

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Theorem 4.20. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have 

∞

e−t t a−1 0 Supersin1 (−; c; zt)dt

0

 1  (c) t a−1 (1 − t)c−a−1 0 Supersin0 (−; −; λzt)dt =  (c − a) 0  1  (c) t a−1 (1 − t)c−a−1 sin (λzt)dt. =  (c − a) 0

(4.73)

Theorem 4.21. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have 

∞

e−t t a−1 0 Supercos 1 (−; c; zt)dt

0

 1  (c) t a−1 (1 − t)c−a−1 0 Supercos 0 (−; −; λzt)dt  (c − a) 0  1  (c) t a−1 (1 − t)c−a−1 cos (λzt)dt. =  (c − a) 0

=

(4.74)

Property 4.3. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have 1 Supersin1 (a; c; z)

 1  (c) = t a−1 (1 − t)c−a−1 0 Supersin0 (−; −; zt)dt  (a)  (c − a) 0  1  (c) t a−1 (1 − t)c−a−1 sin (zt) dt =  (a)  (c − a) 0

(4.75)

and 1 Supersin1 (a; c; λz)

 1  (c) t a−1 (1 − t)c−a−1 0 Supersin0 (−; −; λzt)dt  (a)  (c − a) 0  1  (c) t a−1 (1 − t)c−a−1 sin (λzt) dt. =  (a)  (c − a) 0

=

(4.76)

Property 4.4. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have 1 Supercos 1 (a; c; z)

 1  (c) t a−1 (1 − t)c−a−1 0 Supercos 0 (−; −; zt)dt  (a)  (c − a) 0  1  (c) t a−1 (1 − t)c−a−1 cos (zt) dt =  (a)  (c − a) 0

=

(4.77)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

251

and 1 Supercos 1 (a; c; λz)

 1  (c) t a−1 (1 − t)c−a−1 0 Supercos 0 (−; −; λzt)dt  (a)  (c − a) 0  1  (c) t a−1 (1 − t)c−a−1 cos (λzt) dt. =  (a)  (c − a) 0

=

(4.78)

Theorem 4.22. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, |z| < 1, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supersin1 a; c; λt κ dt B (α, β) 0  (4.79) ; a, ακ , · · · , α+κ−1 κ α+β−1 κ =z λz × 1+κ Supersin1+κ , α+β+κ−1 ; c, α+β κ ,··· , κ where λ is a constant. Theorem 4.23. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, |z| < 1, and Re (c − a) > 0, then we have  z

 1 t α−1 (z − t)β−1 1 Supersin1 a; c; λt 2 dt B (α, β) 0  (4.80) a, α2 , α+1 2 ; α+β−1 2 =z λz × 3 Supersin3 , α+β+1 c, α+β ; 2 , 2 where λ is a constant. Theorem 4.24. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, and Re (c − a) > 0, then we have  1

 1 t α−1 (1 − t)β−1 1 Supersin1 a; c; λt 2 dt B (α, β) 0  (4.81) a, α2 , α+1 2 ; = 3 Supersin3 λ , α+β+1 c, α+β ; 2 , 2 where λ is a constant. Theorem 4.25. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, |z| < 1, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supersin1 a; c; λ (z − t)κ dt = zα+β−1 B (α, β) 0  (4.82) a, βκ , · · · , β+κ−1 ; κ κ × 1+κ F1+κ λz , α+β+κ−1 ; c, α+β κ ,··· , κ where λ is a constant.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 4.26. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, |z| < 1, and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supersin1 a; c; λt κ (z − t)s dt = zα+β−1 B (α, β) 0  α (4.83) , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s × 1+κ+s Supersin1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c, α+β κ+s , · · · , κ+s where λ is a constant. Theorem 4.27 (Laplace transform). Let a, c, z ∈ C and Re (s) > Re (z). Then the Laplace transform of the function t λ−1 1 Supersin1 (a; c; zt) is given as follows:

L t λ−1 1 Supersin1 (a; c; zt)  ∞ e−st 1 Supersin1 (a; c; zt)dt = (4.84) 0    (λ) λ, a z = λ 2 Supersin1 ; , c s s where the Laplace transform is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(4.85)

0

Theorem 4.28. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, |z| < 1, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supercos 1 a; c; λt κ dt B (α, β) 0  (4.86) ; a, ακ , · · · , α+κ−1 κ α+β−1 κ =z λz × 1+κ Supercos 1+κ , α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant. Theorem 4.29. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, |z| < 1, and Re (c − a) > 0, then we have  z

 1 t α−1 (z − t)β−1 1 Supercos 1 a; c; λt 2 dt B (α, β) 0  (4.87) a, α2 , α+1 2 ; α+β−1 2 =z × 3 Supercos 3 , α+β+1 λz ; c, α+β 2 , 2 where λ is a constant.

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

253

Theorem 4.30. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, and Re (c − a) > 0, then we have  1

 1 t α−1 (1 − t)β−1 1 Supercos 1 a; c; λt 2 dt B (α, β) 0  (4.88) a, α2 , α+1 2 ; = 3 Supercos 3 α+β+1 λ , c, α+β ; 2 , 2 where λ is a constant. Theorem 4.31. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, |z| < 1, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supercos 1 a; c; λ (z − t)κ dt = zα+β−1 B (α, β) 0  (4.89) a, βκ , · · · , β+κ−1 ; κ κ λz × 1+κ Supercos 1+κ , α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant. Theorem 4.32. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, |z| < 1, and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supercos 1 a; c; λt κ (z − t)s dt = zα+β−1 B (α, β) 0  α (4.90) , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s × 1+κ+s Supercos 1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c, α+β κ+s , · · · , κ+s where λ is a constant. Theorem 4.33 (Laplace transform). Let a, c, z ∈ C and Re (s) > Re (z). Then the Laplace transform of the function t λ−1 1 Supercos 1 (a; c; zt) is given as follows:

L t λ−1 1 Supercos 1 (a; c; zt)  ∞ = e−st 1 Supercos 1 (a; c; zt)dt (4.91) 0    (λ) λ, a z , = λ 2 Supercos 1 ; c s s where the Laplace transform is  L [f (t)] = f (s) = 0

∞

e−st f (t) dt.

(4.92)

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At the moment, we derive the following relations: 1 Supersin1 (a; c; −z) = −1 Supersin1 (a; c; z),

(4.93)

1 Supercos 1 (a; c; −z) = 1 Supercos 1 (a; c; z) ,

(4.94)

1 Supertan1 (a; c; −z) = −1 Supertan1 (a; c; z),

(4.95)

1 Supercot 1 (a; c; −z) = −1 Supercot 1 (a; c; z),

(4.96)

1 Supersec 1 (a; c; −z) = 1 Supersec 1 (a; c; z) ,

(4.97)

1 Supercosec 1 (a; c; −z) = −1 Supercosec 1 (a; c; z).

(4.98)

and

4.3 The hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of first type In this section, based on the Kummer confluent hypergeometric series of first type, we propose the hypergeometric superhyperbolic functions, present the integral representation for the hypergeometric superhyperbolic functions, and show the Laplace transforms for the hypergeometric superhyperbolic sine and cosine.

4.3.1 Definitions and properties for the hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of first type We now present the definitions and properties for the hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of first type. Definition 4.9. The hypergeometric superhyperbolic sine via Kummer confluent hypergeometric series of first type is defined as 1 Supersinh1 (a; c; z) =

∞  (a)2n+1 n=0

(c)2n+1

z2n+1 , (2n + 1)!

(4.99)

where a, c ∈ C. Definition 4.10. The hypergeometric superhyperbolic cosine via Kummer confluent hypergeometric series of first type is defined as 1 Supercosh1 (a; c; z) =

∞  (a)2n z2n , (c)2n (2n)! n=0

where a, c ∈ C.

(4.100)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

255

Definition 4.11. The hypergeometric superhyperbolic tangent via Kummer confluent hypergeometric series of first type is defined as 1 Supertanh1 (a; c; z) =

1 Supersinh1 (a; c; z) 1 Supercosh1 (a; c; z)

(4.101)

,

where a, c ∈ C. Definition 4.12. The hypergeometric superhyperbolic cotangent via Kummer confluent hypergeometric series of first type is defined as 1 Supercoth1 (a; c; z) =

1 Supercosh1 (a; c; z) 1 Supersinh1 (a; c; z)

(4.102)

,

where a, c ∈ C. Definition 4.13. The hypergeometric superhyperbolic secant via Kummer confluent hypergeometric series of first type is defined as 1 Supersech1 (a; c; z) =

1 , 1 Supercosh1 (a; c; z)

(4.103)

where a, c ∈ C. Definition 4.14. The hypergeometric superhyperbolic cosecant via Kummer confluent hypergeometric series of first type is defined as 1 Supercosech1 (a; c; z) =

1 1 Supersinh1 (a; c; z)

,

(4.104)

where a, c ∈ C. In this case, we have 1 Supersinh1 (a; c; λz)

1 (1 F1 (a; c; λz) − 1 F1 (a; c; −λz)) 2 ∞  (a)2n+1 (λz)2n+1 , = (c)2n+1 (2n + 1)! =

(4.105)

n=0

1 Supercosh1 (a; c; λz)

1 (1 F1 (a; c; λz) + 1 F1 (a; c; −λz)) 2 ∞  (a)2n (λz)2n , = (c)2n (2n)! =

(4.106)

n=0

1 Supertanh1 (a; c; λz) =

1 Supersinh1 (a; c; λz) 1 Supercosh1 (a; c; λz)

,

(4.107)

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1 Supercoth1 (a; c; λz) = 1 Supersech1 (a; c; λz) =

1 Supercosh1 (a; c; λz)

,

(4.108)

1 , Supercosh 1 1 (a; c; λz)

(4.109)

1 Supersinh1 (a; c; λz)

1 Supercosech1 (a; c; λz) =

where a, c, λ ∈ C and i =

1 , 1 Supersinh1 (a; c; λz)

(4.110)

√ −1.

Definition 4.15. The hypergeometric superhyperbolic sine 1 Supersinh1 (a; c; z), hypergeometric superhyperbolic cosine 1 Supercosh1 (a; c; z), hypergeometric superhyperbolic tangent 1 Supertanh1 (a; c; z), hypergeometric superhyperbolic cotangent 1 Supercoth1 (a; c; z), hypergeometric superhyperbolic secant 1 Supersech1 (a; c; z), and hypergeometric superhyperbolic cosecant 1 Supercosech1 (a; c; z) are called the hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of first type. As a direct result, we have the following property. Property 4.5. If Re (c) > Re (a) > 0, then we have 1 Supersinh1 (a; c; −z) = −1 Supersinh1 (a; c; z),

(4.111)

1 Supercosh1 (a; c; −z) = 1 Supercosh1 (a; c; z) ,

(4.112)

1 Supertanh1 (a; c; −z) = −1 Supertanh1 (a; c; z),

(4.113)

1 Supercoth1 (a; c; −z) = −1 Supercoth1 (a; c; z),

(4.114)

1 Supersech1 (a; c; −z) = 1 Supersech1 (a; c; z) ,

(4.115)

1 Supercosech1 (a; c; −z) = −1 Supercosech1 (a; c; z).

(4.116)

and

Property 4.6. The hypergeometric series of the form 1 F1 (a; c; iλz) =

∞  (a)n (iλz)n

,

(4.117)

d 2ϕ dϕ − iaλϕ = 0, + (b − iλz) 2 dz dz

(4.118)

n=0

(c)n

n!

is a solution of the differential equation z where a, λ, z ∈ C, i =

√

−1, c ∈ C\Z− 0 , and |z| < 1.

At the moment, we have the relations given by 1 F1 (a; c; λz) = 1 Supercosh1 (a; c; λz) + 1 Supersinh1 (a; c; λz)

(4.119)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

257

and 1 F1 (a; c; −λz) = 1 Supercosh1 (a; c; λz) − 1 Supersinh1 (a; c; λz) ,

(4.120)

where a, c, λ ∈ C.

4.3.2 Integral representation for the hypergeometric superhyperbolic functions We now present the integral representation and differential equations for the hypergeometric superhyperbolic sine and hypergeometric superhyperbolic cosine via Kummer confluent hypergeometric series of first type. Property 4.7. If Re (c) > Re (a) > 0, then we have  ∞ e−t t a−1 0 Supersinh1 (−; c; zt)dt 0

 1  (c) t a−1 (1 − t)c−a−1 0 Supersinh0 (−; −; λzt)dt =  (c − a) 0  1  (c) t a−1 (1 − t)c−a−1 sinh (λzt)dt. =  (c − a) 0

(4.121)

Property 4.8. If Re (c) > Re (a) > 0, then we have  ∞ e−t t a−1 0 Supercosh1 (−; c; zt)dt 0

 1  (c) t a−1 (1 − t)c−a−1 0 Supercosh0 (−; −; λzt)dt =  (c − a) 0  1  (c) t a−1 (1 − t)c−a−1 cosh (λzt)dt. =  (c − a) 0

(4.122)

Theorem 4.34. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have  1  (c) Supersinh t a−1 (1 − t)c−a−1 sinh (λz) dt, (4.123) c; λz) = (a; 1 1  (a)  (c − a) 0 where sinh (λz) =

 1  λzt e − e−λzt . 2

(4.124)

Proof. By means of the integral relations 1 F1 (a; c; λz) =

 (c)  (a)  (c − a)

and 1 F1 (a; c; −λz) =

 (c)  (a)  (c − a)



1

t a−1 (1 − t)c−a−1 eλzt dt

(4.125)

t a−1 (1 − t)c−a−1 e−λzt dt,

(4.126)

0

 0

1

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where λ ∈ C and Re (c) > Re (a), we have 1 Supersinh1 (a; c; λz) ∞  (a)2n+1 (λz)2n+1

=

n=0

(c)2n+1 (2n + 1)!

1 (1 F1 (a; c; λz) − 1 F1 (a; c; −λz)) 2    1    (c) a−1 c−a−1 1 λzt −λzt = e −e dt t (1 − t)  (a)  (c − a) 0 2  1  (c) t a−1 (1 − t)c−a−1 sinh (λz) dt, =  (a)  (c − a) 0 =

where sinh (λz) =

 1  λzt e − e−λzt . 2

(4.127)

(4.128)

Theorem 4.35. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have  1  (c) t a−1 (1 − t)c−a−1 cosh (λz) dt, (4.129) 1 Supercosh1 (a; c; λz) =  (a)  (c − a) 0 where

 1  λz e + e−λz . 2 Proof. By means of the integral relationships  1  (c) t a−1 (1 − t)c−a−1 eλzt dt 1 F1 (a; c; λz) =  (a)  (c − a) 0 cosh (λz) =

and 1 F1 (a; c; −λz) =

 (c)  (a)  (c − a)



1

t a−1 (1 − t)c−a−1 e−λzt dt,

(4.130)

(4.131)

(4.132)

0

where λ ∈ C and Re (c) > Re (a), we have 1 Supercosh1 (a; c; λz) ∞  (a)2n (λz)2n

=

n=0

(c)2n (2n)!

1 (1 F1 (a; c; λz) + 1 F1 (a; c; −λz)) 2    1    (c) a−1 c−a−1 1 λzt −λzt e +e = dt t (1 − t)  (a)  (c − a) 0 2  1  (c) = t a−1 (1 − t)c−a−1 cosh (λz) dt,  (a)  (c − a) 0 =

(4.133)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

259

since  1  λz e + e−λz . 2

cosh (λz) =

(4.134)

Theorem 4.36. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have  ∞ 1 e−t t a−1 0 Supersinh1 (−; c; zt)dt, (4.135) 1 Supersinh1 (a; c; z) =  (a) 0 where 0 Supersinh1 (−; c; z) =

1 (0 F1 (−; c; z) − 0 F1 (−; c; −z)) . 2

(4.136)

Proof. Using the integral relations 1 1 F1 (a; c; z) =  (a) and 1 F1 (a; c; −z) =

1  (a)



∞

e−t t a−1 0 F1 (−; c; z)dt

(4.137)

e−t t a−1 0 F1 (−; c; −z)dt,

(4.138)

0



∞

0

we have 1 Supersin1 (a; c; z) ∞  (a)2n+1 z2n+1

=

n=0

(c)2n+1 (2n + 1)!

1 (1 F1 (a; c; z) − 1 F1 (a; c; −z)) 2i    ∞ 1 1 e−t t a−1 = (0 F1 (−; c; zt) − 0 F1 (−; c; −zt)) dt  (a) 0 2  ∞ 1 e−t t a−1 0 Supersinh1 (−; c; zt)dt, =  (a) 0

=

(4.139)

where 0 Supersinh1 (−; c; z) =

1 (0 F1 (−; c; z) − 0 F1 (−; c; −z)) . 2i

(4.140)

Theorem 4.37. If a, c, z ∈ C, Re (c) > 0, Re (a) > 0, and Re (c) > Re (a), then we have  ∞ 1 Supercosh e−t t a−1 0 Supercosh1 (−; c; zt)dt, c; z) = (4.141) (a; 1 1  (a) 0

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where 0 Supercosh1 (−; c; z) =

1 (0 F1 (−; c; z) + 0 F1 (−; c; −z)) . 2

(4.142)

Proof. In view of the relations 1 1 F1 (a; c; z) =  (a) and 1 F1 (a; c; −z) =

1  (a)



∞

e−t t a−1 0 F1 (−; c; z)dt

(4.143)

e−t t a−1 0 F1 (−; c; −z)dt,

(4.144)

0



∞

0

we have that 1 Supercosh1 (a; c; z) ∞  (a)2n z2n

=

n=0

(c)2n (2n)!

1 (1 F1 (a; c; z) + 1 F1 (a; c; −z)) 2    ∞ 1 −t a−1 1 = e t (0 F1 (−; c; zt) + 0 F1 (−; c; −zt)) dt  (a) 0 2  ∞ 1 = e−t t a−1 0 Supercosh1 (−; c; zt)dt,  (a) 0 =

(4.145)

which is derived from 0 Supercosh1 (−; c; z) =

1 (0 F1 (−; c; z) + 0 F1 (−; c; −z)) . 2

(4.146)

Without proofs, we present the following theorems. Theorem 4.38. If Re (α) > 0, Re (a) > 0, Re (c) > 0, |z| < 1, Re (c − a) > 0, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supersinh1 a; c; λt κ dt B (α, β) 0  (4.147) ; a, ακ , · · · , α+κ−1 κ α+β−1 κ =z λz × 1+κ Supersinh1+κ , α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant.

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

261

Theorem 4.39. If Re (α) > 0, Re (a) > 0, Re (c) > 0, |z| < 1, and Re (c − a) > 0, then we have  z

 1 t α−1 (z − t)β−1 1 Supersinh1 a; c; λt 2 dt B (α, β) 0  (4.148) a, α2 , α+1 2 ; α+β−1 2 =z λz × 3 Supersinh3 , α+β+1 ; c, α+β 2 , 2 where λ is a constant. Theorem 4.40. If Re (α) > 0, Re (a) > 0, Re (c) > 0, and Re (c − a) > 0, then we have  1

 1 t α−1 (1 − t)β−1 1 Supersinh1 a; c; λt 2 dt B (α, β) 0  (4.149) a, α2 , α+1 2 ; = 3 Supersinh3 α+β+1 λ , c, α+β ; 2 , 2 where λ is a constant. Theorem 4.41. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, |z| < 1, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supersinh1 a; c; λ (z − t)κ dt = zα+β−1 B (α, β) 0  (4.150) a, βκ , · · · , β+κ−1 ; κ κ λz , × 1+κ Supersinh1+κ α+β+κ−1 ; c, α+β κ ,··· , κ where λ is a constant. Theorem 4.42. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, |z| < 1, and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supersinh1 a; c; λt κ (z − t)s dt = zα+β−1 B (α, β) 0  α (4.151) , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s × 1+κ+s Supersinh1+κ+s , α+β+κ+s−1 (κ + s)κ+s c, α+β ; κ+s , · · · , κ+s where λ is a constant. Theorem 4.43. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supersinh1 a; c; λt κ dt B (α, β) 0  (4.152) ; a, ακ , · · · , α+κ−1 κ = 1+κ Supersinh1+κ α+β+κ−1 λ , ; c, α+β κ ,··· , κ where λ is a constant.

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Theorem 4.44. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supersinh1 a; c; λ (1 − t)κ dt B (α, β) 0  (4.153) a, βκ , · · · , β+κ−1 ; κ = 1+κ Supersinh1+κ λ , α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant. Theorem 4.45. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ, s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supersinh1 a; c; λt κ (1 − t)s dt B (α, β) 0  α (4.154) , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s = 1+κ+s Supersinh1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c, α+β κ+s , · · · , κ+s where λ is a constant. Theorem 4.46 (Laplace transform). Let a, c, z ∈ C and Re (s) > Re (z). Then the Laplace transform of the function t λ−1 1 Supersinh1 (a; c; zt) is given as follows:

L t λ−1 1 Supersinh1 (a; c; zt)  ∞ e−st 1 Supersinh1 (a; c; zt)dt = (4.155) 0    (λ) λ, a z ; , = λ 2 Supersinh1 c s s where the Laplace transform is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(4.156)

0

Theorem 4.47. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, |z| < 1, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supercosh1 a; c; λt κ dt B (α, β) 0  (4.157) α α+κ−1 a, , · · · , ; κ κ α+β−1 κ =z λz , × 1+κ Supercosh1+κ α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant.

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

263

Theorem 4.48. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, |z| < 1, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supercosh1 a; c; λ (z − t)κ dt = zα+β−1 B (α, β) 0  (4.158) a, βκ , · · · , β+κ−1 ; κ κ × 1+κ Supercosh1+κ λz , α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant. Theorem 4.49. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, |z| < 1, and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Supercosh1 a; c; λt κ (z − t)s dt = zα+β−1 B (α, β) 0  α (4.159) , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s × 1+κ+s Supercosh1+κ+s , α+β+κ+s−1 (κ + s)κ+s c, α+β ; κ+s , · · · , κ+s where λ is a constant. Theorem 4.50. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supercosh1 a; c; λt κ dt B (α, β) 0  (4.160) a, ακ , · · · , α+κ−1 ; κ = 1+κ Supercosh1+κ λ , α+β+κ−1 ; c, α+β κ ,··· , κ where λ is a constant. Theorem 4.51. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supercosh1 a; c; λ (1 − t)κ dt B (α, β) 0  (4.161) a, βκ , · · · , β+κ−1 ; κ = 1+κ Supercosh1+κ λ , α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant. Theorem 4.52. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ, s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supercosh1 a; c; λt κ (1 − t)s dt B (α, β) 0  α (4.162) , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s = 1+κ+s Supercosh1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c, α+β κ+s , · · · , κ+s where λ is a constant.

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Theorem 4.53 (Laplace transform). Let a, c, z ∈ C and Re (s) > Re (z). Then the Laplace transform of the function t λ−1 1 Supercosh1 (a; c; zt) is given as follows:

L t λ−1 1 Supercosh1 (a; c; zt)  ∞ = e−st 1 Supercosh1 (a; c; zt)dt (4.163) 0    (λ) λ, a z ; , = λ 2 Supercosh1 c s s where the Laplace transform is



L [f (t)] = f (s) =

∞

e−st f (t) dt.

(4.164)

0

At the moment, we have the following properties. Property 4.9. Let a, c, λ ∈ C. Then 1 Supersinh1 (a; c; z) = −i 1 Supersin1 (a; c; iz),

(4.165)

1 Supercosh1 (a; c; z) = 1 Supercos 1 (a; c; iz),

(4.166)

1 Supertanh1 (a; c; z) = −i 1 Supertan1 (a; c; iz),

(4.167)

1 Supercoth1 (a; c; z) = i 1 Supercot 1 (a; c; iz),

(4.168)

1 Supersech1 (a; c; z) = 1 Supersec 1 (a; c; iz),

(4.169)

1 Supercosech1 (a; c; z) = i 1 Supercosec 1 (a; c; iz).

(4.170)

and

Property 4.10. Let a, c, λ ∈ C. Then 1 Supersin1 (a; c; −λz) = −1 Supersin1 (a; c; λz) ,

(4.171)

1 Supercos 1 (a; c; −λz) = 1 Supercos 1 (a; c; λz),

(4.172)

1 Supertan1 (a; c; −λz) = −1 Supertan1 (a; c; λz),

(4.173)

1 Supercot 1 (a; c; λz) =1 Supercot 1 (a; c; λz) ,

(4.174)

1 Supersec 1 (a; c; −λz) = 1 Supersec 1 (a; c; λz) ,

(4.175)

1 Supercosec 1 (a; c; −λz) = −1 Supercosec 1 (a; c; λz) .

(4.176)

and

Property 4.11. Let a, c, λ ∈ C. Then 1 Supersinh1 (a; c; −λz) = −1 Supersinh1 (a; c; λz) ,

(4.177)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

265

1 Supercosh1 (a; c; −λz) = 1 Supercosh1 (a; c; λz) ,

(4.178)

1 Supertanh1 (a; c; −λz) = −1 Supertanh1 (a; c; λz) ,

(4.179)

1 Supercoth1 (a; c; −λz) = −1 Supercoth1 (a; c; λz) ,

(4.180)

1 Supersech1 (a; c; −λz) = 1 Supersech1 (a; c; λz) ,

(4.181)

1 Supercosech1 (a; c; −λz) = −1 Supercosech1 (a; c; λz) .

(4.182)

and

4.4 The Kummer confluent hypergeometric series of second type In this section, we present the definition and theorems for the Kummer confluent hypergeometric series of second type.

4.4.1 Definition and theorems for the Kummer confluent hypergeometric series of second type We now begin with the definition for the Kummer confluent hypergeometric series of second type. Definition 4.16. The Kummer confluent hypergeometric series of second type is defined by (see [126], p. 471) 1 K1 (a; c; z)

 (1 − c)  (c − 1) (1 F1 (a; c; z)) + (1 F1 (a − c + 1; 2 − c; z))  (a − c + 1)  (a) ∞ ∞  (1 − c)  (a)n zn  (c − 1)  (a − c + 1)n zn + , =  (a − c + 1)  (a) (c)n n! (2 − c)n n!

=

n=0

(4.183)

n=0

where a, z ∈ C, c ∈ C\Z− 0 , and |z| < 1. Generally, the Kummer confluent hypergeometric series of second type can be written as [126] 1 K1 (a; c; λz)

 (1 − c)  (c − 1) (1 F1 (a; c; λz)) + (1 F1 (a − c + 1; 2 − c; λz))  (a − c + 1)  (a) ∞ ∞  (1 − c)  (a)n (λz)n  (c − 1)  (a − c + 1)n (λz)n = + ,  (a − c + 1)  (a) n! (c)n n! (2 − c)n

=

n=0

where a, λ, z ∈ C, c ∈ C\Z− 0 , and |z| < 1.

n=0

(4.184)

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For more results on the confluent hypergeometric functions, see [127]. Without proofs, we present the following theorems. Theorem 4.54. Let a, z ∈ C, c ∈ C\Z− 0 , |z| < 1, and Re (c) > Re (a). Then we have  1  (c)  (1 − c) t a−1 (1 − t)c−a−1 ezt dt  (a)  (c − a)  (a − c + 1) 0  1  (c − 1)  (2 − c) t a−c (1 − t)−a ezt dt. +  (a)  (1 − a)  (a − c + 1) 0

1 K1 (a; c; z) =

(4.185)

Theorem 4.55. Let a, z ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, and Re (c) > Re (a). Then we have  1  (c)  (1 − c) t a−1 (1 − t)c−a−1 eλzt dt  (a)  (c − a)  (a − c + 1) 0  1  (c − 1)  (2 − c) t a−c (1 − t)−a eλzt dt. +  (a)  (1 − a)  (a − c + 1) 0

1 K1 (a; c; λz) =

(4.186)

Theorem 4.56. Let a, λ, z ∈ C, c ∈ C\Z− 0 , |z| < 1, and Re (c) > Re (a) > 0. Then we have 1 K1 (a; c; λz)

 ∞  (1 − c) = e−t t a−1 0 F1 (−; c; λzt)dt  (a − c + 1)  (a) 0  ∞  (c − 1) e−t t a−c 0 F1 (−; 2 − c; λzt)dt. +  (a)  (a − c + 1) 0

(4.187)

Theorem 4.57. If α, β, z, a ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, Re (c) > Re (a), and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 K1 a; c; λt κ dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  a, ακ , · · · , α+κ−1 ; κ κ λz × 1+κ F1+κ α+β+κ−1 c, α+β ; κ ,··· , κ  (c − 1) α+β−1 ·z +  (a)  ; κ a − c + 1, ακ , · · · , α+κ−1 κ × 1+κ F1+κ , α+β+κ−1 λz 2 − c, α+β ; κ ,··· , κ where λ is a constant.

(4.188)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

267

Theorem 4.58. If α, β, z, a ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, Re (c) > Re (a), and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 K1 a; c; λ (z − t)κ dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  a, βκ , · · · , β+κ−1 ; κ κ λz × 1+κ F1+κ (4.189) α+β+κ−1 c, α+β ; κ ,··· , κ  (c − 1) α+β−1 ·z +  (a)  a − c + 1, βκ , · · · , β+κ−1 ; κ × 1+κ F1+κ λzκ , α+β+κ−1 2 − c, α+β , · · · , ; κ κ where λ is a constant. Theorem 4.59. If α, β, z, a ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, Re (c) > Re (a), and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 K1 a; c; λt κ (z − t)s dt B (α, β) 0  (1 − c) = · zα+β−1  (a − c + 1)  α β β+s−1 κ s s λzκ+s , , · · · , ; a, κ , · · · , α+κ−1 κ κ s s × 1+κ+s F1+κ+s (4.190) α+β+κ+s−1 (κ + s)κ+s c, α+β , · · · , ; κ+s κ+s  (c − 1) α+β−1 ·z  (a)  , βs , · · · , β+s−1 ; κ κ s s λzκ+s a − c + 1, ακ , · · · , α+κ−1 κ s × 1+κ+s F1+κ+s , α+β+κ+s−1 (κ + s)κ+s 2 − c, α+β ; κ+s , · · · , κ+s +

where λ is a constant.

4.5 The hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of second type In this section, we consider the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of second type. We propose the properties and theorems for the hypergeometric supersine and hypergeometric supercosine via Kummer confluent hypergeometric series of second type. We also give the series of the hypergeometric supertrigonometric functions of the Kummer confluent hypergeometric series of second type. Finally, we present the integral representations for the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of second type.

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4.5.1 Definitions and theorems for the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of second type We now present definitions and theorems for the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of second type. Definition 4.17. The hypergeometric supersine via Kummer confluent hypergeometric series of second type is defined as 1 Ksupersin1 (a; c; z) ∞ 

=

 (1 − c)  (a − c + 1)

n=0

∞

(a)2n+1 (−1)n (λz)2n+1  (c − 1)  (a − c + 1)2n+1 (−1)n (λz)2n+1 + ,  (a) (c)2n+1 (2n + 1)! (2 − c)2n+1 (2n + 1)! n=0

(4.191) where a, c ∈ C. Definition 4.18. The hypergeometric supercosine via Kummer confluent hypergeometric series of second type is defined as 1 Ksupercos 1 (a; c; z) ∞ 

=

 (1 − c)  (a − c + 1)

n=0

∞

(a)2n (−1)n z2n  (c − 1)  (a − c + 1)2n (−1)n z2n + ,  (a) (c)2n (2n)! (2 − c)2n (2n)!

(4.192)

n=0

where a, c ∈ C. Definition 4.19. The hypergeometric supertangent via Kummer confluent hypergeometric series of second type is defined as 1 Ksupertan1 (a; c; z) =

1 Ksupersin1 (a; c; z) 1 Ksupercos 1 (a; c; z)

,

(4.193)

where a, c ∈ C. Definition 4.20. The hypergeometric supercotangent via Kummer confluent hypergeometric series of second type is defined as 1 Ksupercot 1 (a; c; z) =

1 Ksupercos 1 (a; c; z) 1 Ksupersin1 (a; c; z)

,

(4.194)

where a, c ∈ C. Definition 4.21. The hypergeometric supersecant via Kummer confluent hypergeometric series of second type is defined as 1 Ksupersec 1 (a; c; z) =

where a, c ∈ C.

1 , Ksupercos 1 1 (a; c; z)

(4.195)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

269

Definition 4.22. The hypergeometric supercosecant via Kummer confluent hypergeometric series of second type is defined as 1 Ksupercosec 1 (a; c; z) =

1 , Ksupersin 1 1 (a; c; z)

(4.196)

where a, c ∈ C. Let us discuss the relations among the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of second type. By means of the series representations 1 K1 (a; c; iλz)

 (1 − c)  (c − 1) (1 F1 (a; c; iλz)) + (1 F1 (a − c + 1; 2 − c; iλz))  (a − c + 1)  (a) ∞ ∞  (1 − c)  (a)n (iλz)n  (c − 1)  (a − c + 1)n (iλz)n = +  (a − c + 1)  (a) n! (c)n n! (2 − c)n

=

n=0

(4.197)

n=0

and 1 K1 (a; c − iλz)

 (1 − c)  (c − 1) (1 F1 (a; c; −iλz)) + (1 F1 (a − c + 1; 2 − c; −iλz))  (a − c + 1)  (a) ∞ ∞  (1 − c)  (a)n (−iλz)n  (c − 1)  (a − c + 1)n (−iλz)n = + ,  (a − c + 1) n!  (a) n! (c)n (2 − c)n

=

n=0

(4.198)

n=0

we define the hypergeometric supersine via Kummer confluent hypergeometric series of second type by 1 Ksupersin1 (a; c; λz)

= = + = =

1 (1 K1 (a; c; iλz) − 1 K1 (a; c; −iλz)) 2i   1  (1 − c) (1 F1 (a; c; iλz) − 1 F1 (a; c; −iλz))  (a − c + 1) 2i    (c − 1) 1 (1 F1 (a − c + 1; 2 − c; iλz) − 1 F1 (a − c + 1; 2 − c; −iλz))  (a) 2i  (1 − c)  (c − 1) 1 Supersin1 (a; c; λz) + 1 Supersin1 (a − c + 1; 2 − c; λz)  (a − c + 1)  (a) ∞ ∞  (1 − c)  (a)2n+1 (−1)n (λz)2n+1  (c − 1)  (a − c + 1)2n+1 (−1)n (λz)2n+1 + ,  (a − c + 1)  (a) (c)2n+1 (2n + 1)! (2 − c)2n+1 (2n + 1)! n=0

n=0

(4.199)

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the hypergeometric supercosine via Kummer confluent hypergeometric series of second type by 1 Ksupercos 1 (a; c; λz)

= = + = =

1 (1 K1 (a; c; iλz) + 1 K1 (a; c; −iλz)) 2    (1 − c) 1 (1 F1 (a; c; iλz) + 1 F1 (a; c; −iλz))  (a − c + 1) 2    (c − 1) 1 (1 F1 (a − c + 1; 2 − c; iλz) + 1 F1 (a − c + 1; 2 − c; −iλz))  (a) 2  (1 − c)  (c − 1) 1 Supercos 1 (a; c; λz) + 1 Supercos 1 (a − c + 1; 2 − c; λz)  (a − c + 1)  (a) ∞ ∞  (1 − c)  (a)2n (−1)n (λz)2n  (c − 1)  (a − c + 1)2n (−1)n (λz)2n + ,  (a − c + 1)  (a) (c)2n (2n)! (2 − c)2n (2n)! n=0

(4.200)

n=0

the hypergeometric supertangent via Kummer confluent hypergeometric series of second type by 1 Ksupertan1 (a; c; λz) =

1 Ksupersin1 (a; c; λz) 1 Ksupercos 1 (a; c; λz)

,

(4.201)

the hypergeometric supercotangent via Kummer confluent hypergeometric series of second type by 1 Ksupercot 1 (a; c; λz) =

1 Ksupercos 1 (a; c; λz) 1 Ksupersin1 (a; c; λz)

,

(4.202)

the hypergeometric supersecant via Kummer confluent hypergeometric series of second type by 1 Ksupersec 1 (a; c; λz) =

1 , 1 Ksupercos 1 (a; c; λz)

(4.203)

and the hypergeometric supercosecant via Kummer confluent hypergeometric series of second type by 1 Ksupercosec 1 (a; c; λz) =

where a, c, λ ∈ C and i =

√ −1.

1 , 1 Ksupersin1 (a; c; λz)

(4.204)

Definition 4.23. The hypergeometric supersine 1 Ksupersin1 (a; c; z), hypergeometric supercosine 1 Ksupercos 1 (a; c; z), hypergeometric supertangent 1 Ksupertan1 (a; c; z), hypergeometric supercotangent 1 Ksupercot 1 (a; c; z), hypergeometric supersecant 1 Ksupersec 1 (a; c; z), and hypergeometric supercosecant 1 Ksupercosec 1 (a; c; z) are called the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of second type.

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

271

Moreover, we have the relation 1 K1 (a; c; iλz) = 1 Ksupercos 1 (a; c; λz) + i 1 Ksupersin1 (a; c; λz)

where a, c, λ ∈ C and i =

(4.205)

√ −1.

4.5.2 The series of the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of second type We now consider the series of the hypergeometric supertrigonometric functions of the Kummer confluent hypergeometric series of second type. Series 1 In this case, we structure series of the form  (a; c; z) =

∞ 

φ (κ) (1 K1 (a; c; iκz)),

(4.206)

κ=0

which can be represented in the form  (a; c; z) = γ +

∞       ϕ (κ) 1 Ksupercos 1 (a; c; κz) + ψ (κ) 1 Ksupersin1 (a; c; κz) , (4.207) κ=1

where ϕ (0) − iψ (0) , 2 ϕ (κ) − iψ (κ) , φ (κ) = 2 γ=

(4.208) (4.209)

and ϕ (κ) + iψ (κ) (4.210) 2 are the coefficients of the hypergeometric supertrigonometric functions of the Kummer confluent hypergeometric series of second type. φ (−κ) =

Series 2 If  (a; c; −z) = − (a; c; z), then there exists the series of the form  (a; c; z) =

∞ 

  ψ (κ) 1 Ksupersin1 (a; c; κz) ,

(4.211)

κ=1

where ψ (κ) are the coefficients of the hypergeometric supertrigonometric functions of the Kummer confluent hypergeometric series of second type.

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Series 3 If  (a; c; −z) =  (a; c; z), then there exists the series of the form  (a; c; z) = γ +

∞ 

  ϕ (κ) 1 Ksupercos 1 (a; c; κz) ,

(4.212)

κ=1

where γ and ϕ (κ) are the coefficients of the hypergeometric supertrigonometric functions of the Kummer confluent hypergeometric series of second type. In this case, we have 1 Ksupersin1 (a; c; −z) = −1 Ksupersin1 (a; c; z),

(4.213)

1 Supercos 1 (a; c; −z) = 1 Ksupercos 1 (a; c; z),

(4.214)

1 Ksupertan1 (a; c; −z) = −1 Ksupertan1 (a; c; z),

(4.215)

1 Ksupercot 1 (a; c; −z) = −1 Ksupercot 1 (a; c; z),

(4.216)

1 Ksupersec 1 (a; c; −z) = 1 Ksupersec 1 (a; c; z),

(4.217)

1 Ksupercosec 1 (a; c; −z) = −1 Ksupercosec 1 (a; c; z).

(4.218)

and

4.5.3 Integral representations for the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of second type We now consider the integral representations for the hypergeometric supertrigonometric functions via Kummer confluent hypergeometric series of second type. Theorem 4.60. Let a, z ∈ C, c ∈ C\Z− 0 , |z| < 1, and Re (c) > Re (a). Then we have 1 Ksupersin1 (a; c; z)

 1  (c)  (1 − c) t a−1 (1 − t)c−a−1 sin (zt) dt  (a)  (c − a)  (a − c + 1) 0  1  (c − 1)  (2 − c) t a−c (1 − t)−a sin (zt) dt. +  (a)  (1 − a)  (a − c + 1) 0 =

(4.219)

Theorem 4.61. Let a, z ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, and Re (c) > Re (a). Then we have 1 Ksupersin1 (a; c; λz)

 1  (c)  (1 − c) t a−1 (1 − t)c−a−1 sin (λzt) dt  (a)  (c − a)  (a − c + 1) 0  1  (c − 1)  (2 − c) t a−c (1 − t)−a sin (λzt) dt. +  (a)  (1 − a)  (a − c + 1) 0 =

(4.220)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

273

Theorem 4.62. Let a, λ, z ∈ C, c ∈ C\Z− 0 , |z| < 1, and Re (c) > Re (a) > 0. Then we have 1 Ksupersin1 (a; c; λz)

 ∞  (1 − c) = e−t t a−1 0 Supersin1 (−; c; λzt)dt  (a − c + 1)  (a) 0  ∞  (c − 1) e−t t a−c 0 Supersin1 (−; 2 − c; λzt)dt. +  (a)  (a − c + 1) 0

(4.221)

Theorem 4.63. If α, β, z, a ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, Re (c) > Re (a), and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Ksupersin1 a; c; λt κ dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  ; a, ακ , · · · , α+κ−1 κ κ × 1+κ Supersin1+κ α+β+κ−1 λz c, α+β ; κ ,··· , κ  (c − 1) α+β−1 ·z +  (a)  ; κ a − c + 1, ακ , · · · , α+κ−1 κ λz , × 1+κ Supersin1+κ α+β+κ−1 2 − c, α+β ; κ ,··· , κ

(4.222)

where λ is a constant. Theorem 4.64. If α, β, z, a ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, Re (c) > Re (a), and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Ksupersin1 a; c; λ (z − t)κ dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  a, βκ , · · · , β+κ−1 ; κ κ × 1+κ Supersin1+κ λz α+β+κ−1 c, α+β ; κ ,··· , κ  (c − 1) α+β−1 ·z +  (a)  a − c + 1, βκ , · · · , β+κ−1 ; κ κ λz , × 1+κ Supersin1+κ α+β+κ−1 2 − c, α+β ; κ ,··· , κ where λ is a constant.

(4.223)

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Theorem 4.65. If α, β, z, a ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, Re (c) > Re (a), and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Ksupersin1 a; c; λt κ (z − t)s dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  α , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s × 1+κ+s Supersin1+κ+s α+β+κ+s−1 (κ + s)κ+s ; c, α+β κ+s , · · · , κ+s  (c − 1) α+β−1 + ·z  (a)  , βs , · · · , β+s−1 ; κ κ s s λzκ+s a − c + 1, ακ , · · · , α+κ−1 κ s × 1+κ+s Supersin1+κ+s κ+s , α+β+κ+s−1 + s) (κ , · · · , ; 2 − c, α+β κ+s κ+s

(4.224)

where λ is a constant. Theorem 4.66. Let a, z ∈ C, c ∈ C\Z− 0 , |z| < 1, and Re (c) > Re (a). Then we have 1 Ksupercos 1 (a; c; z)

 1  (c)  (1 − c) t a−1 (1 − t)c−a−1 cos (zt) dt  (a)  (c − a)  (a − c + 1) 0  1  (c − 1)  (2 − c) t a−c (1 − t)−a cos (zt) dt. +  (a)  (1 − a)  (a − c + 1) 0 =

(4.225)

Theorem 4.67. Let a, z ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, and Re (c) > Re (a). Then we have 1 Ksupercos 1 (a; c; λz)

 1  (c)  (1 − c) t a−1 (1 − t)c−a−1 cos (λzt) dt  (a)  (c − a)  (a − c + 1) 0  1  (c − 1)  (2 − c) t a−c (1 − t)−a cos (λzt) dt. +  (a)  (1 − a)  (a − c + 1) 0 =

(4.226)

Theorem 4.68. Let a, λ, z ∈ C, c ∈ C\Z− 0 , |z| < 1, and Re (c) > Re (a) > 0. Then we have 1 Ksupercos 1 (a; c; λz)

 ∞  (1 − c) e−t t a−1 0 Supercos 1 (−; c; λzt)dt  (a − c + 1)  (a) 0  ∞  (c − 1) e−t t a−c 0 Supercos 1 (−; 2 − c; λzt)dt. +  (a)  (a − c + 1) 0 =

(4.227)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

275

Theorem 4.69. If α, β, z, a ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, Re (c) > Re (a), and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Ksupercos 1 a; c; λt κ dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  ; a, ακ , · · · , α+κ−1 κ κ × 1+κ Supercos 1+κ (4.228) α+β+κ−1 λz ; c, α+β κ ,··· , κ  (c − 1) α+β−1 ·z +  (a)  ; κ a − c + 1, ακ , · · · , α+κ−1 κ × 1+κ Supercos 1+κ , α+β+κ−1 λz ; 2 − c, α+β κ ,··· , κ where λ is a constant. Theorem 4.70. If α, β, z, a ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, Re (c) > Re (a), and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Ksupercos 1 a; c; λ (z − t)κ dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  a, βκ , · · · , β+κ−1 ; κ κ × 1+κ Supercos 1+κ λz (4.229) α+β+κ−1 c, α+β ; κ ,··· , κ  (c − 1) α+β−1 ·z +  (a)  a − c + 1, βκ , · · · , β+κ−1 ; κ λzκ , × 1+κ Supercos 1+κ α+β+κ−1 , · · · , ; 2 − c, α+β κ κ where λ is a constant. Theorem 4.71. If α, β, z, a ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, Re (c) > Re (a), and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Ksupercos 1 a; c; λt κ (z − t)s dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  α , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s × 1+κ+s Supercos 1+κ+s (4.230) α+β+κ+s−1 (κ + s)κ+s c, α+β ; κ+s , · · · , κ+s

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+

 (c − 1) α+β−1 ·z  (a)



× 1+κ+s Supercos 1+κ+s

, βs , · · · , β+s−1 ; κ κ s s λzκ+s a − c + 1, ακ , · · · , α+κ−1 κ s α+β+κ+s−1 (κ + s)κ+s ; 2 − c, α+β κ+s , · · · , κ+s

,

where λ is a constant.

4.6 The hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of second type In this section, we consider the hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of second type.

4.6.1 Definitions and properties for the hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of second type We now consider the definitions of the hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of second type. Definition 4.24. The hypergeometric superhyperbolic supersine via Kummer confluent hypergeometric series of second type is defined as 1 Ksupersinh1 (a; c; z) ∞ 

 (1 − c) =  (a − c + 1)

n=0

∞

 (c − 1)  (a − c + 1)2n+1 z2n+1 (a)2n+1 z2n+1 + ,  (a) (c)2n+1 (2n + 1)! (2 − c)2n+1 (2n + 1)!

(4.231)

n=0

where a, c ∈ C. Definition 4.25. The hypergeometric superhyperbolic supercosine via Kummer confluent hypergeometric series of second type is defined as 1 Ksupercosh1 (a; c; z) ∞ 

 (1 − c) =  (a − c + 1)

n=0

∞

 (c − 1)  (a − c + 1)2n z2n (a)2n z2n + ,  (a) (c)2n (2n)! (2 − c)2n (2n)!

(4.232)

n=0

where a, c ∈ C. Definition 4.26. The hypergeometric superhyperbolic supertangent via Kummer confluent hypergeometric series of second type is defined as 1 Ksupertanh1 (a; c; λz) =

where a, c ∈ C.

1 Ksupersinh1 (a; c; λz) 1 Ksupercosh1 (a; c; λz)

,

(4.233)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

277

Definition 4.27. The hypergeometric superhyperbolic supercotangent via Kummer confluent hypergeometric series of second type is defined as 1 Ksupercoth1 (a; c; λz) =

1 Ksupercosh1 (a; c; λz) 1 Ksupersinh1 (a; c; λz)

,

(4.234)

where a, c ∈ C. Definition 4.28. The hypergeometric superhyperbolic supersecant via Kummer confluent hypergeometric series of second type is defined as 1 Ksupersech1 (a; c; λz) =

1 , 1 Ksupercosh1 (a; c; λz)

(4.235)

where a, c ∈ C. Definition 4.29. The hypergeometric superhyperbolic supercosecant via Kummer confluent hypergeometric series of second type is defined as 1 Ksupercosech1 (a; c; λz) =

1 , Ksupersinh 1 1 (a; c; λz)

(4.236)

where a, c ∈ C. In this case, we have the following relations among the hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of second type. Using 1 K1 (a; c; λz)

 (1 − c)  (c − 1) (1 F1 (a; c; λz)) + (1 F1 (a − c + 1; 2 − c; λz))  (a − c + 1)  (a) ∞ ∞  (1 − c)  (a)n (λz)n  (c − 1)  (a − c + 1)n (λz)n = +  (a − c + 1)  (a) n! (c)n n! (2 − c)n

=

n=0

(4.237)

n=0

and 1 K1 (a; c − λz)

 (1 − c)  (c − 1) (1 F1 (a; c; −λz)) + (1 F1 (a − c + 1; 2 − c; −λz))  (a − c + 1)  (a) ∞ ∞  (1 − c)  (a)n (−λz)n  (c − 1)  (a − c + 1)n (−λz)n = + ,  (a − c + 1)  (a) n! (c)n n! (2 − c)n

=

n=0

n=0

(4.238)

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we have the hypergeometric superhyperbolic supersine via Kummer confluent hypergeometric series of second type 1 Ksupersinh1 (a; c; λz)

= = + = =

1 (1 K1 (a; c; λz) − 1 K1 (a; c; −λz)) 2   1  (1 − c) (1 F1 (a; c; λz) − 1 F1 (a; c; −λz))  (a − c + 1) 2    (c − 1) 1 (4.239) (1 F1 (a − c + 1; 2 − c; λz) − 1 F1 (a − c + 1; 2 − c; −λz))  (a) 2  (1 − c)  (c − 1) 1 Supersinh1 (a; c; λz) + 1 Supersinh1 (a − c + 1; 2 − c; λz)  (a − c + 1)  (a) ∞ ∞  (1 − c)  (a)2n+1 (λz)2n+1  (c − 1)  (a − c + 1)2n+1 (λz)2n+1 + ,  (a − c + 1)  (a) (c)2n+1 (2n + 1)! (2 − c)2n+1 (2n + 1)! n=0

n=0

the hypergeometric superhyperbolic supercosine via Kummer confluent hypergeometric series of second type 1 Ksupercosh1 (a; c; λz)

= = + = =

1 (1 K1 (a; c; λz) + 1 K1 (a; c; −λz)) 2    (1 − c) 1 (1 F1 (a; c; λz) + 1 F1 (a; c; −λz))  (a − c + 1) 2    (c − 1) 1 (4.240) (1 F1 (a − c + 1; 2 − c; λz) + 1 F1 (a − c + 1; 2 − c; −λz))  (a) 2  (1 − c)  (c − 1) 1 Supercosh1 (a; c; λz) + 1 Supercosh1 (a − c + 1; 2 − c; λz)  (a − c + 1)  (a) ∞ ∞  (1 − c)  (a)2n (λz)2n  (c − 1)  (a − c + 1)2n (λz)2n + ,  (a − c + 1)  (a) (c)2n (2n)! (2 − c)2n (2n)! n=0

n=0

the hypergeometric superhyperbolic supertangent via Kummer confluent hypergeometric series of second type 1 Ksupertanh1 (a; c; λz) =

1 Ksupersinh1 (a; c; λz) 1 Ksupercosh1 (a; c; λz)

,

(4.241)

the hypergeometric superhyperbolic supercotangent via Kummer confluent hypergeometric series of second type 1 Ksupercoth1 (a; c; λz) =

1 Ksupercosh1 (a; c; λz) 1 Ksupersinh1 (a; c; λz)

,

(4.242)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

279

the hypergeometric superhyperbolic supersecant via Kummer confluent hypergeometric series of second type 1 Ksupersech1 (a; c; λz) =

1 , 1 Ksupercosh1 (a; c; λz)

(4.243)

and the hypergeometric superhyperbolic supercosecant via Kummer confluent hypergeometric series of second type 1 Ksupercosech1 (a; c; λz) =

1 , Ksupersinh 1 1 (a; c; λz)

(4.244)

where a, c, λ ∈ C. Definition 4.30. The hypergeometric superhyperbolic sine 1 Ksupersinh1 (a; c; z), hypergeometric superhyperbolic cosine hypergeometric 1 Ksupercosh1 (a; c; z), superhyperbolic tangent 1 Ksupertanh1 (a; c; z), hypergeometric superhyperbolic cotangent 1 Ksupercoth1 (a; c; z), hypergeometric superhyperbolic secant 1 Ksupersech1 (a; c; z), and hypergeometric superhyperbolic cosecant 1 Ksupercosech1 (a; c; z) are called the hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of second style. In this case, we have the following property. Property 4.12. 1 Ksupersinh1 (a; c; −z) = −1 Ksupersinh1 (a; c; z),

(4.245)

1 Supercosh1 (a; c; −z) = 1 Ksupercosh1 (a; c; z),

(4.246)

1 Ksupertanh1 (a; c; −z) = −1 Ksupertanh1 (a; c; z),

(4.247)

1 Ksupercoth1 (a; c; −z) = −1 Ksupercoth1 (a; c; z),

(4.248)

1 Ksupersech1 (a; c; −z) = 1 Ksupersech1 (a; c; z),

(4.249)

1 Ksupercosech1 (a; c; −z) = −1 Ksupercosech1 (a; c; z).

(4.250)

and

Moreover, we have that 1 K1 (a; c; λz) = 1 Ksupercosh1 (a; c; λz) + 1 Ksupersinh1 (a; c; λz) ,

(4.251)

where a, c, λ ∈ C. In this case, we present the following property. Property 4.13. 1 Ksupersinh1 (a; c; z) = −i 1 Ksupersin1 (a; c; iz),

(4.252)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

1 Ksupercosh1 (a; c; z) = 1 Ksupercos 1 (a; c; iz),

(4.253)

1 Ksupertanh1 (a; c; z) = −i 1 Ksupertan1 (a; c; iz),

(4.254)

1 Ksupercoth1 (a; c; z) = i 1 Ksupercot 1 (a; c; iz),

(4.255)

1 Ksupersech1 (a; c; z) = 1 Ksupersec 1 (a; c; iz),

(4.256)

1 Ksupercosech1 (a; c; z) = i 1 Ksupercosec 1 (a; c; iz),

(4.257)

and

where a, c, λ ∈ C.

4.6.2 Integral reforestations for hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of second type In this part, we consider the integral reforestations for hypergeometric superhyperbolic functions via Kummer confluent hypergeometric series of second type. Theorem 4.72. Let a, z ∈ C, c ∈ C\Z− 0 , |z| < 1, and Re (c) > Re (a). Then we have 1 Ksupersinh1 (a; c; z)

 1  (c)  (1 − c) t a−1 (1 − t)c−a−1 sinh (zt) dt  (a)  (c − a)  (a − c + 1) 0  1  (c − 1)  (2 − c) t a−c (1 − t)−a sinh (zt) dt. +  (a)  (1 − a)  (a − c + 1) 0 =

(4.258)

Theorem 4.73. Let a, z ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, and Re (c) > Re (a). Then we have 1 Ksupersinh1 (a; c; λz)

 1  (c)  (1 − c) = t a−1 (1 − t)c−a−1 sinh (λzt) dt  (a)  (c − a)  (a − c + 1) 0  1  (c − 1)  (2 − c) t a−c (1 − t)−a sinh (λzt) dt. +  (a)  (1 − a)  (a − c + 1) 0

(4.259)

Theorem 4.74. Let a, λ, z ∈ C, c ∈ C\Z− 0 , |z| < 1, and Re (c) > Re (a) > 0. Then we have 1 Ksupersinh1 (a; c; λz)

 ∞  (1 − c) e−t t a−1 0 Supersinh1 (−; c; λzt)dt  (a − c + 1)  (a) 0  ∞  (c − 1) e−t t a−c 0 Supersinh1 (−; 2 − c; λzt)dt. +  (a)  (a − c + 1) 0 =

(4.260)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

281

Theorem 4.75. If Re (α) > 0, Re (a) > 0, Re (c) > 0, |z| < 1, Re (c − a) > 0 and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Ksupersinh1 a; c; λt κ dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  a, ακ , · · · , α+κ−1 ; κ κ × 1+κ Supersinh1+κ (4.261) α+β+κ−1 λz c, α+β , · · · , ; κ κ  (c − 1) α+β−1 ·z +  (a)  ; κ a − c + 1, ακ , · · · , α+κ−1 κ × 1+κ Supersinh1+κ , α+β+κ−1 λz ; 2 − c, α+β κ ,··· , κ where λ is a constant. Theorem 4.76. If Re (α) > 0, Re (a) > 0, Re (c) > 0, |z| < 1, Re (c − a) > 0, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Ksupersinh1 a; c; λ (z − t)κ dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  a, βκ , · · · , β+κ−1 ; κ κ λz × 1+κ Supersinh1+κ (4.262) α+β+κ−1 c, α+β ; κ ,··· , κ  (c − 1) α+β−1 ·z +  (a)  a − c + 1, βκ , · · · , β+κ−1 ; κ × 1+κ Supersinh1+κ λzκ , α+β+κ−1 2 − c, α+β , · · · , ; κ κ where λ is a constant. Theorem 4.77. If Re (α) > 0, Re (β) > 0, κ, s ∈ N, Re (c − a) > 0, and |z| < 1, then we have  z   1 t α−1 (z − t)β−1 1 Ksupersinh1 a; c; λt κ (z − t)s dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  α β β+s−1 κ s s λzκ+s , , · · · , ; a, κ , · · · , α+κ−1 κ κ s s × 1+κ+s Supersinh1+κ+s (4.263) α+β+κ+s−1 (κ + s)κ+s , · · · , ; c, α+β κ+s κ+s +

 (c − 1) α+β−1 ·z  (a)

282

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions  × 1+κ+s Supersinh1+κ+s

, βs , · · · , β+s−1 ; κ κ s s λzκ+s a − c + 1, ακ , · · · , α+κ−1 κ s α+β+κ+s−1 (κ + s)κ+s 2 − c, α+β ; κ+s , · · · , κ+s

,

where λ is a constant. Theorem 4.78. Let a, z ∈ C, c ∈ C\Z− 0 , |z| < 1, and Re (c) > Re (a). Then we have 1 Ksupercosh1 (a; c; z)

 1  (c)  (1 − c) t a−1 (1 − t)c−a−1 cosh (zt) dt  (a)  (c − a)  (a − c + 1) 0  1  (c − 1)  (2 − c) t a−c (1 − t)−a cosh (zt) dt. +  (a)  (1 − a)  (a − c + 1) 0 =

(4.264)

Theorem 4.79. Let a, z ∈ C, c ∈ C\Z− 0 , |z| < 1, λ ∈ C, and Re (c) > Re (a). Then we have 1 Ksupercosh1 (a; c; λz)

 1  (c)  (1 − c) t a−1 (1 − t)c−a−1 cosh (λzt) dt  (a)  (c − a)  (a − c + 1) 0  1  (c − 1)  (2 − c) + t a−c (1 − t)−a cosh (λzt) dt.  (a)  (1 − a)  (a − c + 1) 0 =

(4.265)

Theorem 4.80. Let a, λ, z ∈ C, c ∈ C\Z− 0 , |z| < 1, and Re (c) > Re (a) > 0. Then we have 1 Ksupercosh1 (a; c; λz)

 ∞  (1 − c) e−t t a−1 0 Supercosh1 (−; c; λzt)dt  (a − c + 1)  (a) 0  ∞  (c − 1) e−t t a−c 0 Supercosh1 (−; 2 − c; λzt)dt. +  (a)  (a − c + 1) 0 =

(4.266)

Theorem 4.81. If a, λ, z ∈ C, c ∈ C\Z− 0 , |z| < 1, Re (c) > Re (a) > 0, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Ksupercosh1 a; c; λt κ dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  ; a, ακ , · · · , α+κ−1 κ κ × 1+κ Supercosh1+κ (4.267) α+β+κ−1 λz c, α+β , · · · , ; κ κ  (c − 1) α+β−1 ·z +  (a)  ; κ a − c + 1, ακ , · · · , α+κ−1 κ λz , × 1+κ Supercosh1+κ α+β+κ−1 ; 2 − c, α+β κ ,··· , κ where λ is a constant.

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

283

Theorem 4.82. If a, λ, z ∈ C, c ∈ C\Z− 0 , |z| < 1, Re (c) > Re (a) > 0, and κ ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Ksupercosh1 a; c; λ (z − t)κ dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  a, βκ , · · · , β+κ−1 ; κ κ × 1+κ Supercosh1+κ λz (4.268) α+β+κ−1 ; c, α+β κ ,··· , κ  (c − 1) α+β−1 ·z +  (a)  a − c + 1, βκ , · · · , β+κ−1 ; κ κ × 1+κ Supercosh1+κ λz , α+β+κ−1 2 − c, α+β , · · · , ; κ κ where λ is a constant. Theorem 4.83. If a, λ, z ∈ C, c ∈ C\Z− 0 , |z| < 1, Re (c) > Re (a) > 0, and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Ksupercosh1 a; c; λt κ (z − t)s dt B (α, β) 0  (1 − c) · zα+β−1 =  (a − c + 1)  α , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s × 1+κ+s Supercosh1+κ+s (4.269) α+β+κ+s−1 (κ + s)κ+s ; c, α+β κ+s , · · · , κ+s +

 (c − 1) α+β−1 ·z  (a)



× 1+κ+s Supercosh1+κ+s

, βs , · · · , β+s−1 ; κ κ s s λzκ+s a − c + 1, ακ , · · · , α+κ−1 κ s α+β α+β+κ+s−1 (κ + s)κ+s 2 − c, κ+s , · · · , ; κ+s

,

where λ is a constant. Some elementary examples are as follows (see [25], p. 67; [45]):  z e−t t a−1 dt = a −1 za × 1 F1 (a; a + 1; −z), γ (a, z) =

(4.270)

0 z 1 F1 (a; c; z) = e 1 F1 (c − a; a; z), ∞ 

 (α + 1) zn ,  (1 + n + α) n! n=0   1 2 2z −; a + , ; z F F 2a; 4z) = e 0 1 1 1 (a; 2 0 F1 (−; a; z) =

and

(4.271) (4.272)

(4.273)

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ez = 1 F1 (a; a; z),

(4.274)

where γ (a, z) is the incomplete gamma function. Moreover, we present the following formulations: 1 Supersin1 (a; a; z) = sin (z) ,

(4.275)

1 Supercos 1 (a; a; z) = cos (z) ,

(4.276)

1 Supertan1 (a; a; z) = tan (a; a; z) ,

(4.277)

1 Supercot 1 (a; a; z) = cot (a; a; z) ,

(4.278)

1 Supersec 1 (a; a; λz) = sec (z) ,

(4.279)

1 Supercosec 1 (a; a; z) = cosec (z) ,

(4.280)

1 Supersinh1 (a; a; z) = sinh (z) ,

(4.281)

1 Supercosh1 (a; a; z) = cosh (z) ,

(4.282)

1 Supertanh1 (a; a; z) = tanh (z) ,

(4.283)

1 Supercoth1 (a; a; z) = coth (z) ,

(4.284)

1 Supersech1 (a; a; z) = sech (z) ,

(4.285)

1 Supercosech1 (a; a; z) = cosech (z) .

(4.286)

and

4.7 Analytic number theory via Kummer confluent hypergeometric series In this section, we investigate the analytic number theory involving the Kummer confluent hypergeometric series. Theorem 4.84. If Re (α) > 0, Re (β) > 0, and λ is a constant, then we have    1 1 α; α−1 β−1 t sin (λt) dt = 1 Supersin1 λ , (1 − t) α + β; B (α, β) 0    1 1 α; λ , t α−1 (1 − t)β−1 cos (λt) dt = 1 Supercos 1 α + β; B (α, β) 0    1 1 α; t α−1 (1 − t)β−1 sinh (λt) dt = 1 Supersinh1 λ , α + β; B (α, β) 0 and 1 B (α, β)





1

t 0

α−1

(1 − t)

β−1

cosh (λt) dt = 1 Supercosh1

 α; λ . α + β;

(4.287) (4.288) (4.289)

(4.290)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

285

Proof. By means of the integral relations    1 1 α; t α−1 (1 − t)β−1 eiλt dt = 1 F1 iλ , α + β; B (α, β) 0    1 1 α; t α−1 (1 − t)β−1 e−iλt dt = 1 F1 − λi , α + β; B (α, β) 0    1 1 α; α−1 β−1 λt t e dt = 1 F1 λ , (1 − t) α + β; B (α, β) 0 and 1 B (α, β)



1

t 0

α−1

(1 − t)

β−1 −λt

e

 dt = 1 F1

 α; −λ , α + β;

(4.291) (4.292) (4.293)

(4.294)

we get the results. Theorem 4.85. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 F1 a; c; λt κ dt B (α, β) 0  (4.295) a, ακ , · · · , α+κ−1 ; κ = 1+κ F1+κ λ , α+β+κ−1 ; c, α+β κ ,··· , κ where λ is a constant. Theorem 4.86. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 F1 a; c; λ (1 − t)κ dt B (α, β) 0  (4.296) a, βκ , · · · , β+κ−1 ; κ = 1+κ F1+κ λ , α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant. Theorem 4.87. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ, s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 F1 a; c; λt κ (1 − t)s dt B (α, β) 0  α (4.297) , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s = 1+κ+s F1+κ+s , α+β+κ+s−1 (κ + s)κ+s c, α+β ; κ+s , · · · , κ+s where λ is a constant.

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Theorem 4.88. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supersin1 a; c; λt κ dt B (α, β) 0  (4.298) ; a, ακ , · · · , α+κ−1 κ = 1+κ F1+κ λ , α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant. Theorem 4.89. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supersin1 a; c; λ (1 − t)κ dt B (α, β) 0  (4.299) a, βκ , · · · , β+κ−1 ; κ = 1+κ Supersin1+κ λ , α+β+κ−1 c, α+β ; κ ,··· , κ where λ is a constant. Theorem 4.90. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ, s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supersin1 a; c; λt κ (1 − t)s dt B (α, β) 0  α (4.300) , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s = 1+κ+s Supersin1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c, α+β κ+s , · · · , κ+s where λ is a constant. Theorem 4.91. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supercos 1 a; c; λt κ dt B (α, β) 0  (4.301) ; a, ακ , · · · , α+κ−1 κ = 1+κ F1+κ α+β+κ−1 λ , ; c, α+β κ ,··· , κ where λ is a constant. Theorem 4.92. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supercos 1 a; c; λ (1 − t)κ dt B (α, β) 0  (4.302) a, βκ , · · · , β+κ−1 ; κ = 1+κ F1+κ λ , α+β+κ−1 ; c, α+β κ ,··· , κ where λ is a constant.

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

287

Theorem 4.93. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, Re (c − a) > 0, and κ, s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Supercos 1 a; c; λt κ (1 − t)s dt B (α, β) 0  α (4.303) , βs , · · · , β+s−1 ; κ κ s s λzκ+s a, κ , · · · , α+κ−1 κ s = 1+κ+s F1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c, α+β κ+s , · · · , κ+s where λ is a constant. Theorem 4.94. If Re (α) > 0, Re (β) > 0, Re (a) > 0, Re (c) > 0, |z| < 1, and Re (c − a) > 0, then we have  z

 1 t α−1 (z − t)β−1 1 Supersinh1 a; c; λt 2 dt B (α, β) 0  (4.304) α α+1 a, , ; 2 2 α+β−1 2 =z × 3 Supersinh3 α+β+1 λz , c, α+β ; 2 , 2 where λ is a constant. Theorem 4.95. If Re (α) > 0, Re (a) > 0, Re (c − a) > 0, Re (c) > 0, and Re (β) > 0, then we have  1

 1 t α−1 (1 − t)β−1 1 Supersinh1 a; c; λt 2 dt B (α, β) 0  (4.305) a, α2 , α+1 2 ; = 3 Supersinh3 λ , α+β+1 c, α+β ; 2 , 2 where λ is a constant. Theorem 4.96. If Re (α) > 0, Re (a) > 0, Re (c − a) > 0, Re (c) > 0, and Re (β) > 0, then we have  z

 1 t α−1 (z − t)β−1 1 Supercosh1 a; c; λt 2 dt B (α, β) 0  (4.306) a, α2 , α+1 2 ; α+β−1 2 =z λz × 3 Supercosh3 , α+β+1 ; c, α+β 2 , 2 where λ is a constant. Theorem 4.97. If Re (α) > 0, Re (a) > 0, Re (c − a) > 0, Re (c) > 0, and Re (β) > 0, then we have  1

 1 t α−1 (1 − t)β−1 1 Supercosh1 a; c; λt 2 dt B (α, β) 0  (4.307) a, α2 , α+1 2 ; = 3 Supercosh3 α+β+1 λ , c, α+β ; 2 , 2 where λ is a constant.

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Theorem 4.98. If Re (α) > 0, Re (a) > 0, and Re (β) > 0, then we have 

 t α−1 (1 − t)β−1 sin λt 2 dt 0  a, ακ , α+1 2 ; = 3 Supersin3 α+β+1 λ , ; a, α+β 2 , κ

1 B (α, β)

1

(4.308)

where λ is a constant. Theorem 4.99. If Re (α) > 0, Re (a) > 0, and Re (β) > 0, then we have 



t α−1 (1 − t)β−1 sin λ (1 − t)2 dt 0  a, β2 , β+1 2 ; = 3 Supersin3 λ , α+β+1 ; a, α+β 2 , 2

1 B (α, β)

1

(4.309)

where λ is a constant. Theorem 4.100. If Re (α) > 0, Re (a) > 0, and Re (β) > 0, then we have 

 t α−1 (1 − t)β−1 cos λt 2 dt 0  a, ακ , α+1 2 ; = 3 Supercos 3 λ , α+β+1 a, α+β ; 2 , κ

1 B (α, β)

1

(4.310)

where λ is a constant. Theorem 4.101. If Re (α) > 0, Re (a) > 0, and Re (β) > 0, then we have 



t α−1 (1 − t)β−1 cos λ (1 − t)2 dt 0  a, β2 , β+1 2 ; = 3 Supercos 3 λ , α+β+1 ; a, α+β 2 , 2

1 B (α, β)

1

(4.311)

where λ is a constant. Theorem 4.102. If Re (α) > 0, Re (a) > 0, and Re (β) > 0, then we have 

 t α−1 (1 − t)β−1 sinh λt 2 dt 0  a, ακ , α+1 2 ; = 3 Supersinh3 λ , α+β+1 a, α+β ; 2 , κ

1 B (α, β)

where λ is a constant.

1

(4.312)

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

289

Theorem 4.103. If Re (α) > 0, Re (a) > 0, and Re (β) > 0, then we have 



t α−1 (1 − t)β−1 sinh λ (1 − t)2 dt 0  a, β2 , β+1 2 ; = 3 Supersinh3 λ , α+β+1 a, α+β ; 2 , 2

1 B (α, β)

1

(4.313)

where λ is a constant. Theorem 4.104. If Re (α) > 0, Re (a) > 0, and Re (β) > 0, then we have 

 t α−1 (1 − t)β−1 cosh λt 2 dt 0  a, ακ , α+1 2 ; = 3 Supercosh3 α+β+1 λ , a, α+β ; 2 , κ

1 B (α, β)

1

(4.314)

where λ is a constant. Theorem 4.105. If Re (α) > 0, Re (a) > 0, and Re (β) > 0, then we have 



t α−1 (1 − t)β−1 cosh λ (1 − t)2 dt 0  a, β2 , β+1 2 ; = 3 Supercosh3 λ , α+β+1 ; a, α+β 2 , 2

1 B (α, β)

1

(4.315)

where λ is a constant. Without the proofs, we present the zeros of the special functions. Theorem 4.106. 1 Supersin1 (a; c; 0) = 0,

(4.316)

1 Supercos 1 (a; c; 0) = 1,

(4.317)

1 Supertan1 (a; c; 0) = 0,

(4.318)

where Re (a) > 0 and Re (c) > 0. Theorem 4.107.

where Re (a) > 0 and Re (c) > 0. Theorem 4.108.

where Re (a) > 0 and Re (c) > 0.

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Theorem 4.109. 1 Supersec 1 (a; c; 0) = 1,

(4.319)

1 Supersinh1 (a; c; 0) = 0,

(4.320)

1 Supercosh1 (a; c; 0) = 1,

(4.321)

1 Supertanh1 (a; c; 0) = 0,

(4.322)

1 Supersech1 (a; c; 0) = 1,

(4.323)

1 Ksupersin1 (a; c; 0) = 0,

(4.324)

1 Ksupercos 1 (a; c; 0) = 1,

(4.325)

1 Ksupertan1 (a; c; 0) = 0,

(4.326)

1 Ksupersec 1 (a; c; 0) = 1,

(4.327)

where Re (a) > 0 and Re (c) > 0. Theorem 4.110.

where Re (a) > 0 and Re (c) > 0. Theorem 4.111.

where Re (a) > 0 and Re (c) > 0. Theorem 4.112.

where Re (a) > 0 and Re (c) > 0. Theorem 4.113.

where Re (a) > 0 and Re (c) > 0. Theorem 4.114.

where Re (a) > 0 and Re (c) > 0. Theorem 4.115.

where Re (a) > 0 and Re (c) > 0. Theorem 4.116.

where Re (a) > 0 and Re (c) > 0. Theorem 4.117.

where Re (a) > 0 and Re (c) > 0.

Chapter 4 • Hypergeometric supertrigonometric and superhyperbolic functions

291

Theorem 4.118. 1 Ksupersinh1 (a; c; 0) = 0,

(4.328)

1 Ksupercosh1 (a; c; 0) = 1,

(4.329)

1 Ksupertanh1 (a; c; 0) = 0,

(4.330)

1 Ksupersech1 (a; c; 0) = 1,

(4.331)

where Re (a) > 0 and Re (c) > 0. Theorem 4.119.

where Re (a) > 0 and Re (c) > 0. Theorem 4.120.

where Re (a) > 0 and Re (c) > 0. Theorem 4.121.

where Re (a) > 0 and Re (c) > 0. For more information of the Kummer confluent hypergeometric series and other Kummer confluent hypergeometric series, see Abad and Sesma (1995) [128], Airey (1926) [129], Webb and Airey (1918) [130], Arfken (1985) [131], Buchholz (1969) [132], Humbert (1920) [133], Iyanaga and Kawada (1980) [134], Koepf (1998) [135], Magnus and Oberhettinger (1948) [136], Morse and Feshbach (1953) [137], Slater (1960) [138], Tricomi (1960) [139], Chaudhry et al. (2004) [140], Erdélyi (1940) [141], Georgiev and Georgieva-Grosse (2003) [142], Barnard et al. (2009) [143], Ancarani and Gasaneo (2008) [144], Miller and Mocanu (1990) [145], Kim et al. (2012) [146], Abadir (1993) [147], Wimp (1965) [148], Sitnik and Mehrez (2016) [149], Saxena et al. (2004) [150], Virchenko (2006) [151], Ponnusamy and Vuorinen (1998) [152], Silverstone et al. (1985) [153], Kimura (1997) [154], Nath (1951) [155], and Zhong et al. (1985) [156].

5 Hypergeometric supertrigonometric and superhyperbolic functions via Jacobi polynomials 5.1 Jacobi polynomials In this section we introduce the Jacobi’s discovery of the polynomials [157,158], which are called the Jacobi polynomials, and then extend the polynomials based on the Gauss hypergeometric series and considered in the Szegö notation [159].

5.1.1 Definition, properties, and theorems for the Jacobi polynomials Definition 5.1. [Jacobi (1826)] (α,β) The Jacobi polynomial of degree n, denoted by Pn (z), is defined as [157,158] Pn(α,β) (z) =

n  (1 + α)n (1 + β)n (z − 1)k (z + 1)n−k k=0

2n k! (n − k)! (1 + α)k (1 + β)n−k

,

(5.1)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . These polynomials were introduced by Jacobi in 1826 [157,158]. Theorem 5.1 (Szegö theorem). [Szegö (1939)] Let Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . Then we have    1−z (1 + α)n −n, n + α + β + 1; α + 1; . F Pn(α,β) (z) = 2 1 n! 2

(5.2)

The result was obtained by Szegö in 1939 [159]. Theorem 5.2 (Luke theorem). [Luke (1969)] Let Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . Then we have    1+z (−1)n (1 + β)n (α,β) , Pn (z) = 2 F1 −n, n + α + β + 1; β + 1; n! 2

(5.3)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . The result was obtained by Luke in 1969 [14]. An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions https://doi.org/10.1016/B978-0-12-824154-7.00011-8 Copyright © 2021 Elsevier Inc. All rights reserved.

293

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Theorem 5.3 (Rodrigues formula). [Rodrigues (1816)] The Rodrigues formula for the Jacobi polynomial series is as follows: Pn(α,β) (z) =

 (−1)n (1 − z)−α (1 + z)−β d n  (1 − z)n+α (1 + z)n+β 2n n! dzn

(5.4)

 (z − 1)−α (1 + z)−β d n  (z − 1)n+α (1 + z)n+β , n n 2 n! dz

(5.5)

and Pn(α,β) (z) =

where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . The result was obtained by Rodrigues in 1816 [160] and reported by Szegö in 1939 [159]. Theorem 5.4. [Szegö (1939)] If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then Pn(α,α) (z) =

1 (α+1,β+1) (z) . (n + α + β + 1) Pn−1 2

(5.6)

The result was obtained by Szegö in 1939 [159]. For α = β = 0, formula (5.1) was reported in the Rodrigues’ paper [160]. Formula (5.1) was independently obtained in 1812 by Ivory [161] and Jacobi [157,162]. For more information, see [163–165]. Theorem 5.5. We have [159]



1



α,− 2 (α,α) P2n (z) = Pn

(z) ,

(5.7)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . The result was obtained by Szegö in 1939 [159]. Some elementary examples are as follows [159]: 

Pn 

Pn



Pn



Pn 1 1 2,2



Pn and

− 12 ,− 12



Pn

(cos z)

− 12 ,− 12





= cos (nz) ,

(5.8)

sin ((n + 1) z) , (n + 1) sin (z)

(5.9)

(1)

(cos z)

1 1 2,2

1 1 2 ,− 2





=

(1)



− 12 , 12

 (cos z) 

(1)

=

sin

  n + 12 z

. sin 2z

(5.10)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

295

Let  (z) be a measure, and let [159,166]  1  (z)1 = | (z)| (1 − z)α (1 + z)β dz < ∞.

(5.11)

Property 5.1 (The Fourier–Jacobi series). Let  1 | (z)| (1 − z)α (1 + z)β dz < ∞.  (z)1 =

(5.12)

−1

−1

Then we have [166]  (z) = where

 ϕ (κ) =

due to



∞  ϕ (κ) (α,β) k=1 k

(α,β)

1

−1

 (z)

Pk

(α,β)

Pk

(z)

(α,β) Pk (1)

(α,β)

(z) Pk

(1 − z)α (1 + z)β dz

1

−1

(5.13)

(1),

Pp(α,β) (z) Pq(α,β) (1) (1 − z)α (1 + z)β dz =

0 (p = q) (α,β)

k

(p = q)

(5.14)

(5.15)

with (α,β)

k

=

2α+β+1  (k + β + 1)  (k + α + 1) .  (2k + α + β + 1)  (k + α + β + 1)  (k + 1)

(5.16)

The above formulae were also presented by Luke in the monograph [14, p. 277] and the book [167]. Property 5.2. For Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , we have Pn(α,β) (−z) = (−1)n Pn(α,β) (z) .

(5.17)

The result was obtained by Szegö in 1939 [159]. Theorem 5.6. We have (1 + α)n , n! where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . Pn(α,β) (1) =

(5.18)

The result was obtained by Szegö in 1939 [159] and reported in [45]. Theorem 5.7 (Rainville theorem). [Rainville (1960)] We have Pn(α,β) (−z) = (−1)n Pn(β,α) (z) , where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 .

(5.19)

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The result was obtained by Rainville in 1960 [45]. Theorem 5.8 (Rainville theorem). [Rainville (1960)] We have Pn(α,β) (−1) = (−1)n

(1 + α)n , n!

(5.20)

where Re (α) > −1, Re (β) > −1, and n ∈ N0 . The result was obtained by Rainville in 1960 [45]. Theorem 5.9 (Bateman theorem). [Bateman (1905)] We have      ∞ (α,β) t k Pk (z) t (1 − z) t (1 + z) −; 1 + α; −; 1 + β; = F F , 0 1 0 1 2 2 (1 + α)n (1 + β)n

(5.21)

k=0

where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . The result was obtained by Bateman in 1905 [168] and reported by Rainville in 1960 [45, p. 269]. Theorem 5.10. [Rainville (1960)] We have ∞ (α,β)  (β + α + 1)n t k P (z) k

k=0

= (1 − t)

(1 + α)n −(β+α+1)



2 F1

 1 1 2t (z − 1) , (β + α + 1) , (β + α + 2) ; 1 + α; 2 2 (1 − t)2

(5.22)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . The result was obtained by Rainville in 1960 [45, p. 269]. Theorem 5.11. [Szegö (1939)] We have (α,α)

P2n

(z)





  (2n + α + 1)  (n + 1) α,− 12  2 Pn = 2z − 1  (2n)  (n + α + 1)    − 12 ,α  n  (2n + α + 1)  (n + 1) Pn 1 − 2z2 = (−1)  (2n + 1)  (n + α + 1)

(5.23)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

297

and (α,α) P2n+1 (z)





 α, 12   (2n + α + 2)  (n + 1) zPn = 2z2 − 1  (2n + 2)  (n + α + 1)   1   2 ,α n  (2n + α + 2)  (n + 1) zPn 1 − 2z2 , = (−1)  (2n + 2)  (n + α + 1)

(5.24)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . The result was obtained by Szegö in 1939 [159]. Detailed discussions of the formulae were considered in Luke’s book [14, p. 274]. Theorem 5.12. [Szegö (1939)] (α,β) The Jacobi polynomials u (z) = Pn (z) are the solutions of the following linear homogeneous differential equation of the second order [159]:

or

  d 2u du + n (n + α + β + 1) u = 0 1 − z2 + (β − α − (α + β + 2) z) 2 dz dz

(5.25)

 d α+1 β+1 du + n (n + α + β + 1) (1 − z)α (1 + z)β u = 0, (1 + z) (1 − z) dz dz

(5.26)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . The result was obtained by Szegö in 1939 [159]. An alternative equation is given as [159] 

1 − z2

 d 2 dz2

+ (β − α − (α + β + 2) z)

d + (n + 1) (n + α + β) = 0 dz

(5.27)

with the solution of the form (z) = (1 − z)α (1 + z)β Pn(α,β) (z) ,

(5.28)

where |z| < 1. Theorem 5.13 (Jacobi theorem). [Jacobi (1859)] If Re (α) > −1, Re (β) > −1, −1 < z < 1, and k ∈ N0 , then the Jacobi generating function is ∞ 

(α,β)

Pk

(z) t k = 2α+β ς −1 (1 − t + ς)−α (1 + t + ς)−β ,

(5.29)

k=0

where 1  2 ς = 1 − 2zt + t 2 .

(5.30)

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The result for the generating function for the Jacobi polynomial is given originally by Jacobi [158], and its proofs were given by Szegö [169], Carlitz [170], and Pólya and Szegö [171]. Theorem 5.14 (Carlitz theorem). [Carlitz (1961)] If Re (α) > −1, Re (β) > −1, −1 < z < 1, and k ∈ N0 , then the generating function for the Jacobi polynomials is ∞  k=0

α (α,−1) P (z) t k = 2α (1 − t + ς) , n+α k

where

1  2 ς = 1 − 2zt + t 2 .

(5.31)

(5.32)

The result was obtained by Carlitz in 1961 [172]. Theorem 5.15. [Rainville (1960)] If Re (α) > −1, Re (β) > −1, −1 < z < 1, and k ∈ N0 , then the generating function for the Jacobi polynomials is ∞  (α + β + 1)k k=0

(α + 1)k −α−β−1

= (1 − t)

(α,β)

Pk

(z) t k

 α+β+1 2 F1

2

 , α+β+1 2t (z − 1) 2 ; . (1 − t)2 α+1

(5.33)

The result was obtained by Rainville in 1960 [45]. Theorem 5.16. [Carlitz (1961)] If Re (α) > −1, Re (β) > −1, −1 < z < 1, and k ∈ N0 , then (α,β) Pk (z) = 2n

  n   n+α n+β k=0

n−k

k

(z − 1)k (z + 1)n−k .

(5.34)

This is an explicit formula for the Jacobi polynomials proposed by Carlitz [172] and reported by Chihara [173]. For detailed considerations, see the Luke’s monograph [14, p. 275]. Theorem 5.17. [Chihara (1978)] If Re (α) > −1, Re (β) > −1, −1 < z < 1, and k ∈ N0 , then 2n (n + α + β) (2n + α + β − 2) Pn(α,β) (z) =   (α,β) (2n + α + β − 1) (2n + α + β) (2n + α + β − 2) + α 2 − β 2 Pn−1 (z) (α,β)

− 2 (n + α + β − 1) (n + β − 1) (2n + α + β) Pn−2 (z) .

(5.35)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

299

The result is the recurrence relations for the Jacobi polynomials proposed by [173]. Theorem 5.18. If Re (α) > −1, Re (β) > −1, −1 < z < 1, m < n, and m, n ∈ N0 , then 2m

∂ m (α,β) (α+m,β+m) P (z) = (n + α + β + 1)n Pn−m (z) . ∂zm n

(5.36)

For the result reported, see the Luke’s monograph [14, p. 276]. Theorem 5.19. [Luke (1969)] If Re (α) > −1, Re (β) > −1, and n ∈ N0 , then 

1

−1

(1 + t)c (1 − t)d Pn(α,β) (t) dt =

(−1)n 2c+d−1 (1 + β)n B (c, d) n!

(5.37)

× 3 F2 (−n, n + α + β + 1, c; β + 1, c + d; 1). Proof. We have the integral representation given by 

1 −1

(1 + t)c (1 − t)d Pn(α,β) (t) dt

  1−z dt (1 + t) (1 − t) 2 F1 −n, n + α + β + 1; α + 1; 2 −1    1+z (−1)n (1 + β)n 1 dt = (1 + t)c (1 − t)d 2 F1 −n, n + α + β + 1; β + 1; n! 2 −1 (1 + α)n = n!

=



1

c

d

(5.38)

(−1)n 2c+d−1 (1 + β)n B (c, d) 3 F2 (−n, n + α + β + 1, c; β + 1, c + d; 1), n!

which can be derived by the expressions Pn(α,β) (z)

   1−z (1 + α)n = 2 F1 −n, n + α + β + 1; α + 1; n! 2   n 1+z (−1) (1 + β)n = 2 F1 −n, n + α + β + 1; β + 1; n! 2

(5.39)

and 1 B (c, d)



1

(1 + t)c (1 − t)d 2 F1 (−n, n + α + β + 1; β + 1; λ (1 + t))dt

−1 = 2c+d−1 3 F2 (−n, n + α

+ β + 1, c; β + 1, c + d; 2λ),

where Re (α) > −1, Re (β) > −1, and n ∈ N0 . The result was reported by Luke [14].

(5.40)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 5.20. [Luke (1969)] If Re (α) > −1, Re (β) > −1, m > n, and m, n ∈ N0 , then zm =

m 

An Pn(α,β) (z),

(5.41)

n=0

where An =

2n m! (n + α + β − 1) 3 F2 (n − m, n + α + 1, c; 2n + α + β − 1; 2) . (m − n)! (2n + α + β − 1)

(5.42)

The result was reported by Luke [14]. Theorem 5.21. If Re (α) > −1, Re (β) > −1, m > n, and m, n ∈ N0 , then z2m+γ =

m 

(α,α)

Mn P2n+γ (z),

(5.43)

n=0

where Mn =

 + γ  (2n + 2α + γ + 1) .  (m − n)! (2n + α + γ + 1)  m + n + α + γ + 32

 (2m + γ )! 2n + α + 22m+2α+γ

1 2

(5.44)

The result was reported by Luke [14]. Theorem 5.22. If Re (α) > −1, Re (β) > −1, m > n, and m, n ∈ N0 , then (1 − z)m =

m 

(α,β)

℘n P2n

(z),

(5.45)

n=0

where Mn =

(−1)n 2m m! (α + m + 1) (2n + α + β − 1)  (n + 1) . (m − n)! (n + α + 1)  (m + n + γ + 1)

(5.46)

The result was reported by Luke [14].

5.1.2 Hypergeometric supertrigonometric functions via Jacobi polynomials In this section, we propose the hypergeometric supertrigonometric functions via Jacobi polynomials. Apparently, we have that   1 + iλz (1 + α)n −n, n + α + β + 1; α + 1; (5.47) Pn(α,β) (−iλz) = F 2 1 n! 2

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

and Pn(α,β) (iλz) =

  1 − iλz (1 + α)n −n, n + α + β + 1; α + 1; , F 2 1 n! 2

301

(5.48)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 . Using the above formulas, we easily set up the definitions of the special functions. Definition 5.2. The hypergeometric supersine via Jacobi polynomials is defined as (1 + α)n 1 Psinn(α,β) (z) = n! 2i      1 + iz 1 − iz − 2 F1 −n, n + α + β + 1; α + 1; , × 2 F1 −n, n + α + β + 1; α + 1; 2 2 (5.49) where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . Definition 5.3. The hypergeometric supercosine via Jacobi polynomials is defined as (1 + α)n 1 Pcos n(α,β) (z) = n! 2      1 + iz 1 − iz + 2 F1 −n, n + α + β + 1; α + 1; , × 2 F1 −n, n + α + β + 1; α + 1; 2 2 (5.50) where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . Definition 5.4. The hypergeometric supertangent via Jacobi polynomials is defined as (α,β)

Ptann(α,β) (z) =

Psinn

(z)

(α,β) Pcos n (z)

(5.51)

,

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Definition 5.5. The hypergeometric supercotangent via Jacobi polynomials is defined as (α,β)

Pcotann(α,β) (z) =

Pcos n

(z)

(α,β) Psinn (z)

,

(5.52)

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Definition 5.6. The hypergeometric supersecant via Jacobi polynomials is defined as Psec n(α,β) (z) =

1 (α,β) Pcos n (z)

where Re (α) > −1, Re (β) > −1, and −1 < z < 1.

,

(5.53)

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Definition 5.7. The hypergeometric supercosecant via Jacobi polynomials is defined as Pcosec n(α,β) (z) =

1 (α,β) Psinn (z)

(5.54)

,

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. At the moment, we show that Pn(α,β) (iz) = Pcos n(α,β) (z) + iPsinn(α,β) (z)

(5.55)

Pn(α,β) (−iz) = Pcos n(α,β) (z) − iPsinn(α,β) (z) ,

(5.56)

and

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Thus we have that Psinn(α,β) (λz)  1  (α,β) Pn = (iλz) − Pn(α,β) (−iλz) 2i   1 (1 + α)n 1 + iλz = 2 F1 −n, n + α + β + 1; α + 1; 2i n! 2   1 − iλz 1 (1 + α)n , − 2 F1 −n, n + α + β + 1; α + 1; 2i n! 2

(5.57)

Pcos (α,β) (λz) n  1  (α,β) Pn = (iλz) + Pn(α,β) (−iλz) 2   1 + iλz 1 (1 + α)n −n, n + α + β + 1; α + 1; = F 2 1 2 n! 2   1 − iλz 1 (1 + α)n −n, n + α + β + 1; α + 1; , + F 2 1 2 n! 2

(5.58)

Ptann(α,β) (λz) (α,β)

=

Psinn 

=

(λz)

(α,β) Pcos n (λz) 2 F1 2 F1



−n, n + α + β + 1; α + 1;

1+iλz 2

−n, n + α + β + 1; α + 1; 1−iλz 2

 ,

(5.59)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

303

Pcotann(α,β) (λz) (α,β)

=

Pcos n 

=

(λz)

(α,β) Psinn (λz) 2 F1



2 F1

−n, n + α + β + 1; α + 1;

1−iλz 2

−n, n + α + β + 1; α + 1; 1+iλz 2

Psec n(α,β) (λz) =

1 (α,β) Pcos n (λz)

(5.60)

 ,

(5.61)

,

and Pcosec n(α,β) (λz) = where i =

1 (α,β) Psinn (λz)

(5.62)

,

√ −1, Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 . (α,β)

Definition 5.8. The hypergeometric supersine Psinn (z), hypergeometric supercosine (α,β) (α,β) Pcos n (z), hypergeometric supertangent Ptann (z), hypergeometric supercotangent (α,β) (α,β) Pcotann (z), hypergeometric supersecant Psec n (z), and hypergeometric supercose(α,β) cant Pcosec n (z) are called the hypergeometric supertrigonometric functions via Jacobi polynomials. Based on the above definitions, it is not difficult to derive the following theorems. √ Theorem 5.23. If α > −1, β > −1, i = −1, t ∈ C, −1 < z < 1, and k ∈ N0 , then we have the following generating functions: ∞ 

(α,β)

Psink

(z) t k

k=0

 2α+β−1  −1 × ς1 (1 − t + ς1 )−α (1 + t + ς1 )−β − ς2−1 (1 − t + ς2 )−α (1 + t + ς2 )−β = i

(5.63)

and ∞ 

(α,β)

Pcos k

k=0

=2

α+β−1

where

and

×

(z) t k



ς1−1 (1 − t

−α

+ ς1 )

−β

(1 + t + ς1 )

+ ς2−1 (1 − t

−α

+ ς2 )

(1 + t + ς2 )

−β

(5.64)

 ,

1  2 ς1 = 1 − 2izt + t 2

(5.65)

1  2 ς2 = 1 + 2izt + t 2 .

(5.66)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Proof. By means of the representations ∞ 

(iz) t k = 2α+β ς1−1 (1 − t + ς1 )−α (1 + t + ς1 )−β

(5.67)

(−iz) t k = 2α+β ς2−1 (1 − t + ς2 )−α (1 + t + ς2 )−β ,

(5.68)

(α,β)

Pk

k=0

and ∞ 

(α,β)

Pk

k=0

where

and

1  2 ς1 = 1 − 2izt + t 2

(5.69)

1  2 ς2 = 1 + 2izt + t 2 ,

(5.70)

we show that ∞ 

(α,β)

Psink

(z) t k

k=0 ∞  1   (α,β) (α,β) Pk (iz) − Pk (−iz) t k 2i

= =

k=0 2α+β−1

(5.71)

 i  × ς1−1 (1 − t + ς1 )−α (1 + t + ς1 )−β − ς2−1 (1 − t + ς2 )−α (1 + t + ς2 )−β and ∞ 

(α,β)

Pcos k

(z) t k

k=0

=

∞  1   (α,β) (α,β) Pk (iz) + Pk (−iz) t k 2

(5.72)

k=0 α+β−1

=2   × ς1−1 (1 − t + ς1 )−α (1 + t + ς1 )−β + ς2−1 (1 − t + ς2 )−α (1 + t + ς2 )−β .

5.1.3 Hypergeometric superhyperbolic functions via Jacobi polynomials In this section, we give the hypergeometric superhyperbolic functions via Jacobi polynomials.

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

305

Now we consider   1 + λz (1 + α)n 2 F1 −n, n + α + β + 1; α + 1; n! 2

(5.73)

  1 − λz (1 + α)n , 2 F1 −n, n + α + β + 1; α + 1; n! 2

(5.74)

Pn(α,β) (−λz) = and Pn(α,β) (λz) =

where Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 . Based on the above formulas, we set up the definitions of the new special functions. Definition 5.9. The hypergeometric superhyperbolic supersine via Jacobi polynomials is defined as (1 + α)n 1 Psinhn(α,β) (z) = n! 2     (5.75)  1+z 1−z − 2 F1 −n, n + α + β + 1; α + 1; , × 2 F1 −n, n + α + β + 1; α + 1; 2 2 where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . Definition 5.10. The hypergeometric superhyperbolic supercosine via Jacobi polynomials is defined as (1 + α)n 1 Pcoshn(α,β) (z) = n! 2      (5.76) 1+z 1−z × 2 F1 −n, n + α + β + 1; α + 1; + 2 F1 −n, n + α + β + 1; α + 1; , 2 2 where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . Definition 5.11. The hypergeometric superhyperbolic supertangent via Jacobi polynomials is defined as (α,β)

Ptanhn(α,β) (z) =

Psinhn

(z)

(α,β) Pcoshn (z)

(5.77)

,

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Definition 5.12. The hypergeometric superhyperbolic supercotangent via Jacobi polynomials is defined as (α,β)

Pcotanhn(α,β) (z) =

Pcoshn

(z)

(α,β) Psinhn (z)

where Re (α) > −1, Re (β) > −1, and −1 < z < 1.

,

(5.78)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Definition 5.13. The hypergeometric superhyperbolic supersecant via Jacobi polynomials is defined as Psechn(α,β) (z) =

1 (α,β) Pcoshn (z)

(5.79)

,

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Definition 5.14. The hypergeometric superhyperbolic supercosecant via Jacobi polynomials is defined as Pcosech(α,β) (z) = n

1 (α,β) Psinhn (z)

,

(5.80)

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Thus we can show that Pn(α,β) (z) = Pcoshn(α,β) (z) + Psinhn(α,β) (z)

(5.81)

Pn(α,β) (z) = Pcoshn(α,β) (z) + Psinhn(α,β) (z) ,

(5.82)

and

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. So we have that Psinh(α,β) (λz) n   1 Pn(α,β) (λz) − Pn(α,β) (−λz) = 2   1 + λz 1 (1 + α)n −n, n + α + β + 1; α + 1; = F 2 1 2 n! 2   1 − λz 1 (1 + α)n , −n, n + α + β + 1; α + 1; − F 2 1 2 n! 2

(5.83)

Pcoshn(α,β) (λz)  1  (α,β) Pn = (λz) + Pn(α,β) (−λz) 2   1 + λz 1 (1 + α)n = F1 −n, n + α + β + 1; α + 1; 2 n! 2 2   1 − λz 1 (1 + α)n −n, n + α + β + 1; α + 1; , + F 2 1 2 n! 2

(5.84)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

307

Ptanhn(α,β) (λz) (α,β)

=

Psinhn

(λz)

(α,β) Pcoshn (λz)

(5.85)

  1+λz 2 F1 −n, n + α + β + 1; α + 1; 2  , = 1−λz 2 F1 −n, n + α + β + 1; α + 1; 2 Pcotanhn(α,β) (λz) (α,β)

=

Pcoshn

(λz)

(α,β) Psinhn (λz)

(5.86)

  1−λz 2 F1 −n, n + α + β + 1; α + 1; 2  , = 1+λz 2 F1 −n, n + α + β + 1; α + 1; 2 Psechn(α,β) (λz) =

1 (α,β) Pcoshn (λz)

(5.87)

,

and Pcosechn(α,β) (λz) =

1 (α,β) Psinhn (λz)

(5.88)

,

√ where i = −1, Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 . Based on the above results, we set up the following definition. (α,β)

Definition 5.15. The hypergeometric superhyperbolic supersine Psinhn (z), hyperge(α,β) ometric superhyperbolic supercosine Pcoshn (z), hypergeometric superhyperbolic su(α,β) (α,β) pertangent Ptanhn (z), hypergeometric superhyperbolic supercotangent Pcotanhn (z), (α,β) hypergeometric superhyperbolic supersecant Psechn (z), and hypergeometric super(α,β) hyperbolic supercosecant Pcosechn (z) are called the hypergeometric superhyperbolic functions via Jacobi polynomials. Based on this definition, we derive the following theorems. Theorem 5.24. If α > −1, β > −1, t ∈ C, −1 < z < 1, and k ∈ N0 , then we have the following generating functions: ∞ 

(α,β)

Psinhk

k=0

=2

α+β−1

×



(z) t k ς3−1 (1 − t

−α

+ ς3 )

−β

(1 + t + ς3 )

− ς4−1 (1 − t

−α

+ ς4 )

−β

(1 + t + ς4 )



(5.89)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and ∞ 

(α,β)

Pcoshk

k=0

=2

α+β−1

×



(z) t k

ς3−1 (1 − t

−α

+ ς3 )

−β

(1 + t + ς3 )

+ ς4−1 (1 − t

−α

+ ς4 )

(1 + t + ς4 )

−β

(5.90)

 ,

where 1  2 ς3 = 1 − 2zt + t 2

(5.91)

1  2 ς4 = 1 + 2zt + t 2 .

(5.92)

and

Proof. Using the expressions ∞ 

(z) t k = 2α+β ς3−1 (1 − t + ς3 )−α (1 + t + ς3 )−β

(5.93)

(−z) t k = 2α+β ς4−1 (1 − t + ς4 )−α (1 + t + ς4 )−β ,

(5.94)

(α,β)

Pk

k=0

and ∞ 

(α,β)

Pk

k=0

where 1  2 ς3 = 1 − 2zt + t 2

(5.95)

1  2 ς4 = 1 + 2zt + t 2 ,

(5.96)

and

we have that ∞ 

(α,β)

Psinhk

(z) t k

k=0

 1   (α,β) (α,β) Pk = (z) − Pk (−z) t k 2 k=0   α+β−1 =2 × ς3−1 (1 − t + ς3 )−α (1 + t + ς3 )−β − ς4−1 (1 − t + ς4 )−α (1 + t + ς4 )−β ∞

(5.97)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

309

and ∞ 

(α,β)

Pcoshk

(z) t k

k=0 ∞  1   (α,β) (5.98) (α,β) Pk (z) + Pk (−z) t k 2 k=0   α+β−1 =2 × ς3−1 (1 − t + ς3 )−α (1 + t + ς3 )−β + ς4−1 (1 − t + ς4 )−α (1 + t + ς4 )−β .

=

For more detail of the Fourier–Jacobi series and the properties of Jacobi polynomial series, see Szegö (1939) [159], Askey (1975) [166], Munot (1969) [174], Darboux (1878) [175], Spencer (2015) [176], Grosswald (2006) [177], Emde and Jahnke (1923) [178], and Grosswald (2015) [179].

5.2 Jacobi–Luke polynomials In this section, we propose the definition and theorems for the Jacobi–Luke polynomials.

5.2.1 Definition and theorems for the Jacobi–Luke polynomials We now consider the following definition for the Jacobi–Luke polynomials [14, p. 276]. Definition 5.16. [Luke (1969)] (α,β) The Jacobi–Luke polynomial of degree n, denoted by n (z), is defined as 

Pn(α,β) (z) (1 + α)n = (2 F1 (−n, n + α + β + 1; α + 1; 1 − z)) n! n (−1) (1 + β)n = (2 F1 (−n, n + α + β + 1; β + 1; z)) , n!

(5.99)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . The result was obtained by Luke in 1969 [14, p. 276]. Theorem 5.25. [Luke (1969)] If Re (α) > −1, Re (β) > −1, 0 < z < 1, and n ∈ N0 , then 

Pn(α,β) (z) = Pn(α,β) (2z − 1) and 

Pn(α,β) (z) = Pn(α,β)



 z+1 . 2

(5.100)

(5.101)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

The result was obtained by Luke in 1969 [14]. Theorem 5.26. If α > −1, β > −1, −1 < z < 1, and k ∈ N0 , then the Luke generating function is ∞ 



Pn(α,β) (z) t k = 2α+β σ −1 (1 − t + σ )−α (1 + t + σ )−β ,

(5.102)

k=0

where 1  2 σ = 1 − 2 (2z − 1) t + t 2 .

(5.103)

Proof. Using the Jacobi generating function, we directly show the result.

5.2.2 Integral representations for the Jacobi–Luke polynomials We now consider integral representations for the Jacobi–Luke polynomials. Theorem 5.27. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

 1 t α−1 (1 − t)β−1 Pn(c,d) λt κ dt B (α, β) 0 (−1)n (1 + d)n = n!   α+κ−1 ; −n, n + c + d + 1, ακ , α+1 κ ,··· , κ λ , × 2+κ F1+κ α+β+1 , · · · , α+β+κ−1 ; d + 1, α+β κ , κ κ

(5.104)

where λ is a constant. Theorem 5.28. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then  0

1



 t α−1 (1 − t)β−1 Pn(c,d) λt κ (1 − t)s dt

(−1)n (1 + d)n × B (α, β) n!   β β+s−1 κ ss λ , , · · · , ; −n, n + c + d + 1, ακ , · · · , α+κ−1 κ κ s s , × 2+κ+s F1+κ+s α+β+κ+s−1 (κ + s)κ+s d + 1, α+β , · · · , ; κ+s κ+s =

where λ is a constant.

(5.105)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

311

Theorem 5.29. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

 t α−1 (1 − t)β−1 Pn(c,d) λ (1 − t)s dt 0

(−1)n (1 + d)n × B (α, β) n!  β+s−1 −n, n + c + d + 1, βs , β+1 ; s ,··· , s λ , × 2+s F1+s α+β+1 , · · · , α+β+s−1 ; d + 1, α+β s , s s =

(5.106)

where λ is a constant. Theorem 5.30. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

 1 (t − 1)α−1 (1 + t)β−1 Pn(c,d) λ (t − 1)κ dt B (α, β) −1 (−1)n (1 + d)n = × 2α+β−1 n!   α+κ−1 ; κ −n, n + c + d + 1, ακ , α+1 κ ,··· , κ λ2 , × 2+κ F1+κ α+β+1 d + 1, α+β , · · · , α+β+κ−1 ; κ , κ κ

(5.107)

where λ is a constant. Theorem 5.31. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then  1

 (t − 1)α−1 (1 + t)β−1 Pn(c,d) λ (t − 1)κ (1 + t)s dt −1

(−1)n (1 + d)n × B (α, β) × 2α+β−1 n!   β β+s−1 κ s s λ2κ+s , , · · · , ; −n, n + c + d + 1, ακ , · · · , α+κ−1 κ κ s s × 2+κ+s F1+κ+s , α+β+κ+s−1 (κ + s)κ+s d + 1, α+β , · · · , ; κ+s κ+s =

(5.108)

where λ is a constant. Theorem 5.32. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

 (t − 1)α−1 (1 + t)β−1 Pn(c,d) λ (1 + t)s dt −1

(−1)n (1 + d)n × B (α, β) × 2α+β−1 n!  β+s−1 −n, n + c + d + 1, βs , β+1 ; s s ,··· , s × 2+s F1+s λ2 , α+β+1 , · · · , α+β+s−1 ; d + 1, α+β s , s s =

where λ is a constant. The results are derived from the particular cases in Chapter 2.

(5.109)

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5.3 Jacobi–Luke-type polynomials In this section, we introduce the definition and theorems for the Jacobi–Luke-type polynomials.

5.3.1 Definition and theorems for the Jacobi–Luke-type polynomials At the moment, we present the definition of the Jacobi–Luke-type polynomials. The Jacobi–Luke-type polynomial of degree n is defined by (α,β) (z) n (1 + α)n = (2 F1 (−n, n + α + β + 1; α + 1; z)) n! n (−1) (1 + β)n = (2 F1 (−n, n + α + β + 1; β + 1; z + 1)) , n!

(5.110)

where Re (α) > −1, Re (β) > −1, 0 < z < 1, and n ∈ N0 . Theorem 5.33. Let Re (α) > −1, Re (β) > −1, 0 < z < 1, and n ∈ N0 . Then we have n(α,β) (z) = Pn(α,β) (1 − 2z) and

 Pn(α,β) (z) = n(α,β)

 1−z . 2

(5.111)

(5.112)

We have the following result. Property 5.3. Let Re (α) > −1, Re (β) > −1, 0 < z < 1, and n ∈ N0 . Then we have      (α,β) (α,β) z + 1 (α,β) 1 − z = n . Pn (z) = Pn 2 2

(5.113)

Theorem 5.34. If α > −1, β > −1, −1 < z < 1, and k ∈ N0 , then the generating function is ∞ 

(α,β)

k

(z) t k = 2α+β ϑ −1 (1 − t + ϑ)−α (1 + t + ϑ)−β ,

(5.114)

k=0

where

 1 2 ϑ = 1 − 2 (1 − 2z) t + t 2 .

(5.115)

Proof. We directly obtain the result from the Jacobi generating function. Theorem 5.35. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  z  (1 + α)n c+d−1 z t c−1 (z − t)d−1 Pn(α,β) (λt) dt = B (c, d) n! 0 × 3 F2 (−n, n + α + β + 1, c; β + 1, c + d; λz),

(5.116)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

and



1 0



t c−1 (1 − t)d−1 Pn(α,β) (λt) dt =

(1 + α)n c+d−1 z B (c, d) n!

313

(5.117)

× 3 F2 (−n, n + α + β + 1, c; β + 1, c + d; λ). As particular cases, we have:  z  (1 + α)n c+d−1 z t c−1 (z − t)d−1 Pn(α,β) (t) dt = B (c, d) n! 0

(5.118)

× 3 F2 (−n, n + α + β + 1, c; β + 1, c + d; z), and

 0

1



t c−1 (1 − t)d−1 Pn(α,β) (t) dt =

(1 + α)n c+d−1 B (c, d) z n!

(5.119)

× 3 F2 (−n, n + α + β + 1, c; β + 1, c + d; 1). (α,β)

Property 5.4. The Jacobi–Luke-type polynomial n (α,β) (z) = n

(z) can be represented as follows:

n  (1 + α)n (1 + β)n t k (1 − t)n−k , 2n k! (n − k)! (1 + α)k (1 + β)n−k

(5.120)

k=0

where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . Theorem 5.36. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then we have n(α,β) (z) = z−α (1 − z)−β

1 d n n+α z (1 − z)n+β . n n! dz

Theorem 5.37. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then we have  1 0 (p = q) , α β (α,β) (α,β) p (z) q (z) θ (1 − z) dz = (α,β) ωp (p = q) , 0

(5.121)

(5.122)

where ωp(α,β) =

 (p + β + 1)  (p + α + 1) .  (2k + α + β + 1)  (p + α + β + 1)  (p + 1)

(5.123)

Theorem 5.38. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then we have ∞  ψ (κ)

 (z) = where

(α,β) k=1 ωk

 ψ (κ) =

(α,β)

1

 (z) 0

(α,β)

k

k

(α,β)

(z) k

(z)

(α,β) k (0)

(z),

zα (1 − z)β dz.

(5.124)

(5.125)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 5.39. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  z (1 + α)n c+d−1 t c−1 (z − t)d−1 n(α,β) (λt) dt = B (c, d) z n! 0

(5.126)

× 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; λz). Proof. Using the expressions n(α,β) (z) = and (see Chapter 2)  z

(1 + α)n 2 F1 (−n, n + α + β + 1; α + 1; z) n!

t c−1 (z − t)d−1 2 F1 (−n, n + α + β + 1; β + 1; λt)dt

0

=z

c+d−1

(5.127)

(5.128)

B (c, d) 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; λz),

we have the integral representation  z t c−1 (z − t)d−1 (α,β) (t) dt n 0  z (1 + α)n t c−1 (z − t)d−1 2 F1 (−n, n + α + β + 1; α + 1; t)dt = n! 0 (1 + α)n c+d−1 z = B (c, d) 3 F2 (−n, n + α + β + 1, c; +1, c + d; z), n!

(5.129)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, c, d ∈ C\Z, and n ∈ N0 . Thus we directly have the following results. Theorem 5.40. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, and n ∈ N0 , then 

1 0

t c−1 (1 − t)d−1 n(α,β) (t) dt =

(1 + α)n B (c, d) n!

(5.130)

× 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; 1). Theorem 5.41. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < λz < 1, λ ∈ R, and n ∈ N0 , then  z (1 + α)n c+d−1 z t c−1 (z − t)d−1 n(α,β) (λt) dt = B (c, d) n! (5.131) 0 × 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; λz). Theorem 5.42. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then  0

1

t c−1 (1 − t)d−1 (α,β) (λt) dt = n

(1 + α)n B (c, d) n!

× 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; λ).

(5.132)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

315

Thus we have the following case: Let Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < z < 1, and n ∈ N0 . The polynomial of degree n ∈ N can be expressed as follows: 

n

α,β c,d



(z) =

(1 + α)n B (c, d) 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; z). n!

(5.133)

Theorem 5.43. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then  0

1

t c−1 (1 − t)d−1 (α,β) (λt) dt = n

(1 + α)n B (c, d) n!

(5.134)

× 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; λ). Theorem 5.44. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < λz < 1, λ ∈ R, and n ∈ N0 , then 

1

−1

(1 + t)c−1 (1 + t)d−1 (α,β) (λ (1 + t)) dt = n

(1 + α)n c+d−1 B (c, d) 2 n!

(5.135)

× 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; 2λ). Theorem 5.45. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then 

  zc+d−1 (1 + α)n B (c, d) t c−1 (z − t)d−1 n(α,β) λt 2 dt = n! 0   c c+1 c+d c+d +1 × 4 F3 −n, n + α + β + 1, , ; α + 1, , ; λz2 . 2 2 2 2 z

(5.136)

Theorem 5.46. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then 

  (1 + α)n B (c, d) λt 2 dt = t c−1 (1 − t)d−1 (α,β) n n! 0   c c+1 c+d c+d +1 × 4 F3 −n, n + α + β + 1, , ; α + 1, , ;λ . 2 2 2 2 1

(5.137)

Theorem 5.47. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then 

  2c+d−1 (1 + α)n B (c, d) (1 + t)c−1 (1 − t)d−1 n(α,β) λ (1 + t)2 dt = n! −1   c c+1 c+d c+d +1 ; α + 1, , ; 4λ . × 4 F3 −n, n + α + β + 1, , 2 2 2 2 1

Theorem 5.48. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  z (1 + α)n c+d−1 z t c−1 (z − t)d−1 (α,β) B (c, d) (λ (z − t)) dt = n n! 0 × 3 F2 (−n, n + α + β + 1, d; α + 1, c + d; λz).

(5.138)

(5.139)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 5.49. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  1 (1 + α)n B (c, d) t c−1 (1 − t)d−1 n(α,β) (λ (1 − t)) dt = n! 0

(5.140)

× 3 F2 (−n, n + α + β + 1, d; α + 1, c + d; λ). Theorem 5.50. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  1 (1 + α)n c+d−1 B (c, d) 2 (1 + t)c−1 (1 − t)d−1 (α,β) (λ (1 − t)) dt = n n! −1

(5.141)

× 3 F2 (−n, n + α + β + 1, d; α + 1, c + d; 2λ). Theorem 5.51. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  z (1 + α)n c+d−1 z t c−1 (z − t)d−1 n(α,β) (λt (z − t)) dt = B (c, d) n! 0   c + d c + d + 1 λz2 × 4 F3 −n, n + α + β + 1, c, d; α + 1, , ; . 2 2 4 Theorem 5.52. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  1 (1 + α)n B (c, d) t c−1 (1 − t)d−1 n(α,β) (λt (1 − t)) dt = n! 0   c+d c+d +1 λ × 4 F3 −n, n + α + β + 1, c, d; α + 1, , ; . 2 2 4 Theorem 5.53. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  1 (1 + α)n c+d−1 2 B (c, d) (1 + t)c−1 (1 − t)d−1 n(α,β) (λ (1 + t) (1 − t)) dt = n! −1   c+d c+d +1 , ;λ . × 4 F3 −n, n + α + β + 1, c, d; α + 1, 2 2

(5.142)

(5.143)

(5.144)

Definition 5.17. [Luke (1969)] The Luke polynomials are defined as (α,β,ν)

(α,β,ν) (z) = n,0 n

(z) = zν (α,β) (z) , n

(5.145)

where (α,β,ν) n,r (z)  z (α,β,ν) = n,r−1 (t) dt 0  z  1 t ν (z − t)r−1 Pn(α,β) (t) dt =  (r) 0

with Re (ν) > −1.

(5.146)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

The result was obtained by Luke in 1969 [14, p. 281]. In fact, we have  z  t ν (z − t)r−1 Pn(α,β) (t) dt 0  (−1)n (1 + β)n z ν = t (z − t)r−1 2 F1 (−n, n + α + β + 1; β + 1; t)dt n! 0 (−1)n (1 + β)n B (ν + 1, r) × zν+r = n! (r) × 3 F2 (−n, n + α + β + 1, ν; β + 1, ν + r + 1; z) ,

317

(5.147)

so that (α,β,ν) (z) n,r  z (α,β,ν) = n,r−1 (t) dt 0  z  1 t ν (z − t)r−1 Pn(α,β) (t) dt =  (r) 0 (−1)n (1 + β)n B (ν + 1, r) × zν+r = n! (r) × 3 F2 (−n, n + α + β + 1, ν; β + 1, ν + r + 1; z).

(5.148)

Thus we have [14] (α,β,ν) n,r (z) =

(−1)n (1 + β)n zν+r n! (ν + 1)r

(5.149)

× 3 F2 (−n, n + α + β + 1, ν; β + 1, ν + r + 1; z), where Re (α) > −1, Re (β) > −1, ν, r ∈ C\Z, −1 < z < 1, λ ∈ R, and n ∈ N0 . The result was obtained by Luke in 1969 [14, p. 281].

5.3.2 Identities for the Jacobi–Luke-type polynomials In this section, we introduce the Laplace transform and identities for the Jacobi–Luke-type polynomials. Theorem 5.54 (Laplace transform for the Jacobi–Luke-type polynomials). Let α, β, λ, z ∈ C, Re (α) > 0, Re (β) > 0, |z| < 1, and t ∈ [0, +∞). (α,β) Then the Laplace transform of the function t λ−1 n (zt) is as follows:   L t λ−1 (α,β) (zt) n   (1 + α)n = L t λ−1 (2 F1 (−n, n + α + β + 1; α + 1; zt)) n!

318

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions  (1 + α)n  λ−1 t2 F1 (−n, n + α + β + 1; α + 1; zt) dt n! 0   (1 + α)n  (λ) λ, −n, n + α + β + 1 z ; , = F 3 1 α+1 n! sλ s 

=

∞

e−st



(5.150)

where the Laplace transform of a function f is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(5.151)

0

Based on the results derived from the particular cases in Chapter 3, we directly present the following theorems without the proofs. Theorem 5.55. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ ∈ N, then  z κ

1 λt dt t α−1 (z − t)β−1 (c,d) n B (α, β) 0 (1 + c)n × zα+β−1 = (5.152) n!   α α+1 α+κ−1 −n, n + c + d + 1, κ , κ , · · · , κ ; κ λz , × 2+κ F1+κ α+β+1 , · · · , α+β+κ−1 ; c + 1, α+β κ , κ κ where λ is a constant. Theorem 5.56. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ, s ∈ N, then  z

t α−1 (z − t)β−1 n(c,d) λt κ (z − t)s dt 0

(1 + c)n × B (α, β) × zα+β−1 n!   , βs , · · · , β+s−1 ; κ κ s s λzκ+s −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s × 2+κ+s F1+κ+s , α+β+κ+s−1 (κ + s)κ+s c + 1, α+β ; κ+s , · · · , κ+s =

(5.153)

where λ is a constant. Theorem 5.57. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ ∈ N, then  z

λ (z − t)s dt t α−1 (z − t)β−1 (c,d) n 0

(1 + c)n × B (α, β) × zα+β−1 n!   β+s−1 −n, n + c + d + 1, βs , β+1 ; s s ,··· , s λz , × 2+s F1+s α+β+1 c + 1, α+β , · · · , α+β+s−1 ; s , s s =

where λ is a constant.

(5.154)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

319

Theorem 5.58. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then 



t α−1 (1 − t)β−1 n(c,d) λt κ dt 0   α+κ−1 ; −n, n + c + d + 1, ακ , α+1 (1 + c)n κ ,··· , κ = × 2+κ F1+κ λ , α+β+1 n! , · · · , α+β+κ−1 ; c + 1, α+β κ , κ κ

1 B (α, β)

1

(5.155)

where λ is a constant. Theorem 5.59. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then  0

1



t α−1 (1 − t)β−1 n(c,d) λt κ (1 − t)s dt

(1 + c)n × B (α, β) n!   , βs , · · · , β+s−1 ; λκ κ s s −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s × 2+κ+s F1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c + 1, α+β κ+s , · · · , κ+s =

(5.156)

where λ is a constant. Theorem 5.60. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  0

1



t α−1 (1 − t)β−1 (c,d) λ (1 − t)s dt n

(1 + c)n × B (α, β) n!   β+s−1 −n, n + c + d + 1, βs , β+1 ; s ,··· , s × 2+s F1+s λ , α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s =

(5.157)

where λ is a constant. Theorem 5.61. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

1 (1 − t)α−1 (1 + t)β−1 n(c,d) λ (1 − t)κ dt B (α, β) −1 (1 + c)n = × 2α+β−1 n!   α+κ−1 ; κ −n, n + c + d + 1, ακ , α+1 κ ,··· , κ λ2 , × 2+κ F1+κ α+β+1 c + 1, α+β , · · · , α+β+κ−1 ; κ , κ κ where λ is a constant.

(5.158)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 5.62. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then 

1

−1



(1 − t)α−1 (1 + t)β−1 n(c,d) λ (t − 1)κ (1 + t)s dt

(1 + c)n × B (α, β) × 2α+β−1 n!   , βs , · · · , β+s−1 ; λκ κ s s 2κ+s −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s × 2+κ+s F1+κ+s , α+β+κ+s−1 (κ + s)κ+s c + 1, α+β ; κ+s , · · · , κ+s =

(5.159)

where λ is a constant. Theorem 5.63. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then 

1

−1



(t − 1)α−1 (1 + t)β−1 n(c,d) λ (1 + t)s dt

(1 + c)n × B (α, β) × 2α+β−1 n!   β+s−1 −n, n + c + d + 1, βs , β+1 , · · · , ; s s λ2s , × 2+s F1+s α+β+1 α+β+s−1 c + 1, α+β , , · · · , ; s s s =

(5.160)

where λ is a constant.

5.3.3 Hypergeometric supertrigonometric functions via Jacobi–Luke-type polynomials In this section, we propose the hypergeometric supertrigonometric functions via Jacobi– Luke-type polynomials. First, we consider n(α,β) (−iλz) =

(1 + α)n 2 F1 (−n, n + α + β + 1; α + 1; −iλz) n!

(5.161)

(1 + α)n 2 F1 (−n, n + α + β + 1; α + 1; iλz), n!

(5.162)

and n(α,β) (iλz) =

where Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 . With the above formulas, we give the definitions for the new special functions. Definition 5.18. The hypergeometric supersine via Jacobi–Luke-type polynomials is defined as (1 + α)n sinn(α,β) (z) = (5.163) 2 Supersin1 (−n, n + α + β + 1; α + 1; z), n! where Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 .

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

321

Definition 5.19. The hypergeometric supercosine via Jacobi–Luke-type polynomials is defined as cos n(α,β) (z) =

(1 + α)n 2 Supercos 1 (−n, n + α + β + 1; α + 1; z), n!

(5.164)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 . Definition 5.20. The hypergeometric supertangent via Jacobi–Luke-type polynomials is defined as (α,β)

tann(α,β) (z) =

sinn

(z)

(α,β) cos n (z)

(5.165)

,

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Definition 5.21. The hypergeometric supercotangent via Jacobi–Luke-type polynomials is defined as (α,β)

cotann(α,β) (z) =

cos n

(z)

(α,β) sinn (z)

(5.166)

,

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Definition 5.22. The hypergeometric supersecant via Jacobi–Luke type polynomials is defined as sec n(α,β) (z) =

1 (α,β) cos n (z)

(5.167)

,

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Definition 5.23. The hypergeometric supercosecant via Jacobi–Luke-type polynomials is defined as cosec n(α,β) (λz) =

1 (α,β) sinn (λz)

,

(5.168)

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Thus we show that n(α,β) (iz) = cos n(α,β) (z) + isinn(α,β) (z)

(5.169)

n(α,β) (−iz) = cos n(α,β) (z) − isinn(α,β) (z) ,

(5.170)

and

where Re (α) > −1, Re (β) > −1, and −1 < z < 1.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

So we have that sinn(α,β) (λz)  1  (α,β) n = (iλz) − n(α,β) (−iλz) 2i 1 (1 + α)n = 2 F1 (−n, n + α + β + 1; α + 1; iλz) 2i n! 1 (1 + α)n − 2 F1 (−n, n + α + β + 1; α + 1; iλz) 2i n! (1 + α)n = 2 Supersin1 (−n, n + α + β + 1; α + 1; λz) , n!

(5.171)

cos n(α,β) (λz)  1  (α,β) n = (iλz) + n(α,β) (−iλz) 2 1 (1 + α)n = 2 F1 (−n, n + α + β + 1; α + 1; iλz) 2 n! 1 (1 + α)n − 2 F1 (−n, n + α + β + 1; α + 1; −iλz) 2 n! (1 + α)n = 2 Supercos 1 (−n, n + α + β + 1; α + 1; λz) , n!

(5.172)

tann(α,β) (λz) (α,β)

=

sinn

(λz)

(α,β) cos n (λz)

2 Supersin1 (−n, n + α

+ β + 1; α + 1; λz) = 2 Supercos 1 (−n, n + α + β + 1; α + 1; λz)

(5.173)

= 2 Supertan1 (−n, n + α + β + 1; α + 1; λz), cotann(α,β) (λz) (α,β)

=

cos n

(λz)

(α,β) sinn (λz)

2 Supercos 1 (−n, n + α

+ β + 1; α + 1; λz) = Supersin n + α + β + 1; α + 1; λz) (−n, 2 1

(5.174)

= 2 Supercotan1 (−n, n + α + β + 1; α + 1; λz), sec n(α,β) (λz) =

1 (α,β) cos n (λz)

(5.175)

,

and cosec n(α,β) (λz) = where i =

1 (α,β) sinn (λz)

,

√ −1, Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 .

(5.176)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

323

(α,β)

Definition 5.24. The hypergeometric supersine sinn (z), hypergeometric supercosine (α,β) (α,β) cos n (z), hypergeometric supertangent tann (z), hypergeometric supercotangent (α,β) (α,β) cotann (z), hypergeometric supersecant sec n (z), and hypergeometric supercose(α,β) cant cosec n (z) are called the hypergeometric supertrigonometric functions via Jacobi– Luke-type polynomials. Based on the above definitions, we derive the following theorems. Theorem 5.64. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  z (1 + α)n c+d−1 t c−1 (z − t)d−1 sinn(α,β) (λt) dt = B (c, d) z n! 0

(5.177)

× 3 Supersin2 (−n, n + α + β + 1, c; α + 1, c + d; λz). Proof. By means of the relations sinn(α,β) (z) = and



z

 1  (α,β) n (iλz) − n(α,β) (−iλz) 2i

t c−1 (z − t)d−1 2 F1 (−n, n + α + β + 1; β + 1; λt)dt

0

(5.178)

(5.179)

= zc+d−1 B (c, d) 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; λz) we show that  z t c−1 (z − t)d−1 sinn(α,β) (λt) dt 0  (1 + α)n z c−1 t = (z − t)d−1 2 Supersin1 (−n, n + α + β + 1; α + 1; λt)dt n! 0 (1 + α)n c+d−1 z = B (c, d) 3 Supersin2 (−n, n + α + β + 1, c; +1, c + d; λz), n!

(5.180)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, c, d ∈ C\Z, and n ∈ N0 . As direct results, we have the following theorems without the proofs. Theorem 5.65. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, and n ∈ N0 , then  z t c−1 (z − t)d−1 sinn(α,β) (t) dt 0

=

(1 + α)n c+d−1 z B (c, d) 3 Supersin2 (−n, n + α + β + 1, c; +1, c + d; z). n!

Theorem 5.66. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, λ ∈ R, and n ∈ N0 , then  1 (1 + α)n B (c, d) t c−1 (1 − t)d−1 sinn(α,β) (t) dt = n! 0 × 3 Supersin2 (−n, n + α + β + 1, c; α + 1, c + d; 1).

(5.181)

(5.182)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 5.67. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then 

1

0

t c−1 (1 − t)d−1 sinn(α,β) (λt) dt =

(1 + α)n B (c, d) n!

(5.183)

× 3 Supersin2 (−n, n + α + β + 1, c; α + 1, c + d; λ). Theorem 5.68. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  z (1 + α)n c+d−1 z t c−1 (z − t)d−1  cosn(α,β) (λt) dt = B (c, d) n! 0

(5.184)

× 3 Supercos 2 (−n, n + α + β + 1, c; α + 1, c + d; λz). Proof. With the relations  cosn(α,β) (z) =

 1  (α,β) n (iλz) + (α,β) (−iλz) n 2

(5.185)

and 

z

t c−1 (z − t)d−1 2 F1 (−n, n + α + β + 1; β + 1; λt)dt

0

(5.186)

= zc+d−1 B (c, d) 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; λz), we obtain  z t c−1 (z − t)d−1 cos n(α,β) (λt) dt 0  (1 + α)n z c−1 t = (z − t)d−1 2 Supercos 1 (−n, n + α + β + 1; α + 1; λt)dt n! 0 (1 + α)n c+d−1 z = B (c, d) 3 Supercos 2 (−n, n + α + β + 1, c; α + 1, c + d; λz), n!

(5.187)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, c, d ∈ C\Z, and n ∈ N0 . Theorem 5.69. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, and n ∈ N0 , then  z t c−1 (z − t)d−1 cos (α,β) (t) dt n 0

(1 + α)n c+d−1 z = B (c, d) 3 Supercos 2 (−n, n + α + β + 1, c; α + 1, c + d; z). n!

(5.188)

Theorem 5.70. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, λ ∈ R, and n ∈ N0 , then 

1 0

t c−1 (1 − t)d−1 cos n(α,β) (t) dt =

(1 + α)n B (c, d) n!

× 3 Supercos 2 (−n, n + α + β + 1, c; α + 1, c + d; 1).

(5.189)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

325

Theorem 5.71. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then  0

1

t c−1 (1 − t)d−1 cos (α,β) (λt) dt = n

(1 + α)n B (c, d) n!

(5.190)

× 3 Supercos 2 (−n, n + α + β + 1, c; α + 1, c + d; λ). Theorem 5.72 (Laplace transform). Let α, β, λ, z ∈ C, Re (α) > 0, Re (β) > 0, |z| < 1, and t ∈ [0, +∞). (α,β) Then the Laplace transform of the function t λ−1 sinn (zt) is as follows:   L t λ−1 sinn(α,β) (zt)

  (1 + α)n  (λ) λ, −n, n + α + β + 1 z , ; = 3 Supersin1 α+1 n! sλ s

(5.191)

where the Laplace transform of a function f is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(5.192)

0

By means of the above definitions we derive the following theorems without the proofs. Theorem 5.73. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ ∈ N, then  z

1 t α−1 (z − t)β−1 sinn(c,d) λt κ dt B (α, β) 0 (1 + c)n × zα+β−1 = n!   −n, n + c + d + 1, ακ , · · · , α+κ−1 ; κ κ λz , × 2+κ Supersin1+κ α+β+κ−1 c + 1, α+β ; κ ,··· , κ

(5.193)

where λ is a constant. Theorem 5.74. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ, s ∈ N, then  0

z



t α−1 (z − t)β−1 sinn(c,d) λt κ (z − t)s dt

(1 + c)n × B (α, β) × zα+β−1 n!   β β+s−1 κ s s λzκ+s , , · · · , ; −n, n + c + d + 1, ακ , · · · , α+κ−1 κ κ s s × 2+κ+s Supersin1+κ+s , α+β+κ+s−1 (κ + s)κ+s , · · · , ; c + 1, α+β κ+s κ+s (5.194) =

where λ is a constant.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 5.75. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ ∈ N, then  z

t α−1 (z − t)β−1 sinn(c,d) λ (z − t)s dt 0

(1 + c)n × B (α, β) × zα+β−1 n!   −n, n + c + d + 1, βs , · · · , β+s−1 ; s × 2+s Supersin1+s λzs , α+β+s−1 c + 1, α+β , · · · , ; s s =

(5.195)

where λ is a constant. Theorem 5.76. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

1 t α−1 (1 − t)β−1 sinn(c,d) λt κ dt B (α, β) 0 (1 + c)n = n!   ; −n, n + c + d + 1, ακ , · · · , α+κ−1 κ λ , × 2+κ Supersin1+κ α+β+κ−1 ; c + 1, α+β κ ,··· , κ

(5.196)

where λ is a constant. Theorem 5.77. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then  0

1



t α−1 (1 − t)β−1 sinn(c,d) λt κ (1 − t)s dt

(1 + c)n × B (α, β) n!   , βs , · · · , β+s−1 ; κ κ ss λ −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s × 2+κ+s Supersin1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c + 1, α+β κ+s , · · · , κ+s (5.197) =

where λ is a constant. Theorem 5.78. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then 

1

0



t α−1 (1 − t)β−1 sinn(c,d) λ (1 − t)s dt

(1 + c)n × B (α, β) n!   −n, n + c + d + 1, βs , · · · , β+s−1 ; s × 2+s Supersin1+s λ , α+β+s−1 ; c + 1, α+β s ,··· , s =

where λ is a constant.

(5.198)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

Theorem 5.79. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

1 λ (t − 1)κ dt (t − 1)α−1 (1 + t)β−1 sin(c,d) n B (α, β) −1 (1 + c)n = × 2α+β−1 n!   ; κ −n, n + c + d + 1, ακ , · · · , α+κ−1 κ λ2 , × 2+κ Supersin1+κ α+β+κ−1 ; c + 1, α+β κ ,··· , κ

327

(5.199)

where λ is a constant. Theorem 5.80. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then  1

(t − 1)α−1 (1 + t)β−1 sinn(c,d) λ (t − 1)κ (1 + t)s dt −1

(1 + c)n × B (α, β) × 2α+β−1 n!   , βs , · · · , β+s−1 ; κ κ s s λ2κ+s −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s × 2+κ+s Supersin1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c + 1, α+β κ+s , · · · , κ+s (5.200) =

where λ is a constant. Theorem 5.81. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

λ (1 + t)s dt (t − 1)α−1 (1 + t)β−1 sin(c,d) n −1

(1 + c)n × B (α, β) × 2α+β−1 n!   −n, n + c + d + 1, βs , · · · , β+s−1 ; s s λ2 , × 2+s Supersin1+s α+β+s−1 c + 1, α+β ; s ,··· , s =

(5.201)

where λ is a constant. Theorem 5.82 (Laplace transform). Let α, β, λ, z ∈ C, Re (α) > 0, Re (β) > 0, |z| < 1, and t ∈ [0, +∞). (α,β) Then the Laplace transform of the function t λ−1 cos n (zt) is as follows:   L t λ−1 cos n(α,β) (zt)   (1 + α)n  (λ) λ, −n, n + α + β + 1 z = ; , Supercos 3 1 α+1 n! sλ s

(5.202)

where the Laplace transform of a function f is  L [f (t)] = f (s) = 0

∞

e−st f (t) dt.

(5.203)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

In view of the above definitions, we present the following theorems without the proofs. Theorem 5.83. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ ∈ N, then  z

1 t α−1 (z − t)β−1 cos n(c,d) λt κ dt B (α, β) 0 (1 + c)n × zα+β−1 = (5.204) n!   α+κ−1 ; κ −n, n + c + d + 1, ακ , α+1 κ ,··· , κ λz , × 2+κ Supercos 1+κ α+β α+β+1 α+β+κ−1 ; c + 1, κ , κ , · · · , κ where λ is a constant. Theorem 5.84. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ, s ∈ N, then  z

t α−1 (z − t)β−1 cos n(c,d) λt κ (z − t)s dt 0

(1 + c)n × B (α, β) × zα+β−1 n!   , βs , · · · , β+s−1 ; κ κ s s λzκ+s −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s × 2+κ+s Supercos 1+κ+s , α+β+κ+s−1 (κ + s)κ+s c + 1, α+β ; κ+s , · · · , κ+s (5.205) =

where λ is a constant. Theorem 5.85. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ ∈ N, then  z

t α−1 (z − t)β−1 cos (c,d) λ (z − t)s dt n 0

(1 + c)n × B (α, β) × zα+β−1 n!   β+s−1 −n, n + c + d + 1, βs , β+1 ; s s ,··· , s × 2+s Supercos 1+s λz , α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s =

(5.206)

where λ is a constant. Theorem 5.86. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

1 t α−1 (1 − t)β−1 cos n(c,d) λt κ dt B (α, β) 0 (1 + c)n = n!   α+κ−1 −n, n + c + d + 1, ακ , α+1 ; κ ,··· , κ λ , × 2+κ Supercos 1+κ α+β+1 c + 1, α+β , · · · , α+β+κ−1 ; κ , κ κ where λ is a constant.

(5.207)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

329

Theorem 5.87. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then  1

t α−1 (1 − t)β−1 cos n(c,d) λt κ (1 − t)s dt 0

=

(1 + c)n × B (α, β) n! 

× 2+κ+s Supercos 1+κ+s

, βs , · · · , β+s−1 ; −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s α+β+κ+s−1 ; c + 1, α+β κ+s , · · · , κ+s

 κ κ ss λ , (κ + s)κ+s (5.208)

where λ is a constant. Theorem 5.88. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

λ (1 − t)s dt t α−1 (1 − t)β−1 cos (c,d) n 0

(1 + c)n × B (α, β) n!   β+s−1 −n, n + c + d + 1, βs , β+1 ; s ,··· , s × 2+s Supercos 1+s λ , α+β+1 c + 1, α+β , · · · , α+β+s−1 ; s , s s =

(5.209)

where λ is a constant. Theorem 5.89. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N , then  1

1 (t − 1)α−1 (1 + t)β−1 cos n(c,d) λ (t − 1)κ dt B (α, β) −1 (−1)n (1 + c)n = × 2α+β−1 n!   α+κ−1 , · · · , ; −n, n + c + d + 1, ακ , α+1 κ κ λ2κ , × 2+κ Supercos 1+κ α+β+1 α+β+κ−1 c + 1, α+β , , · · · , ; κ κ κ

(5.210)

where λ is a constant. Theorem 5.90. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then  1

(t − 1)α−1 (1 + t)β−1 Pcos n(c,d) λ (t − 1)κ (1 + t)s dt −1

(1 + c)n × B (α, β) × 2α+β−1 n!   , βs , · · · , β+s−1 ; κ κ s s λ2κ+s −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s × 2+κ+s Supercos 1+κ+s , α+β+κ+s−1 (κ + s)κ+s c + 1, α+β ; κ+s , · · · , κ+s (5.211) =

where λ is a constant.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 5.91. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

(t − 1)α−1 (1 + t)β−1 cos n(c,d) λ (1 + t)s dt −1

(1 + c)n × B (α, β) × 2α+β−1 n!   β+s−1 −n, n + c + d + 1, βs , β+1 ; s s ,··· , s λ2 , × 2+s Supercos 1+s α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s =

(5.212)

where λ is a constant.

5.3.4 Hypergeometric superhyperbolic functions via Jacobi–Luke-type polynomials In this section, we propose the hypergeometric superhyperbolic functions via Jacobi– Luke-type polynomials. Similarly, we show that n(α,β) (λz) =

(1 + α)n 2 F1 (−n, n + α + β + 1; α + 1; λz) n!

(5.213)

and

(1 + α)n (5.214) 2 F1 (−n, n + α + β + 1; α + 1; −λz), n! where Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 . With the definitions of the special functions, we derive the following theorems. n(α,β) (−λz) =

Definition 5.25. The hypergeometric superhyperbolic supersine via Jacobi–Luke-type polynomials is defined as sinhn(α,β) (z) =

(1 + α)n 2 Supersinh1 (−n, n + α + β + 1; α + 1; z), n!

(5.215)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 . Definition 5.26. The hypergeometric superhyperbolic supercosine via Jacobi–Luke-type polynomials is defined as coshn(α,β) (z) =

(1 + α)n 2 Supercosh1 (−n, n + α + β + 1; α + 1; z), n!

(5.216)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 . Definition 5.27. The hypergeometric superhyperbolic supertangent via Jacobi–Luke-type polynomials is defined as (α,β)

tanhn(α,β) (z) =

sinhn

(z)

(α,β) coshn (z)

where Re (α) > −1, Re (β) > −1, and −1 < z < 1.

,

(5.217)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

331

Definition 5.28. The hypergeometric superhyperbolic supercotangent via Jacobi–Luketype polynomials is defined as (α,β)

cotanhn(α,β) (z) =

coshn

(z)

(α,β) sinhn (z)

(5.218)

,

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Definition 5.29. The hypergeometric superhyperbolic supersecant via Jacobi–Luke-type polynomials is defined as sechn(α,β) (z) =

1 (α,β)

coshn

(5.219)

, (z)

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Definition 5.30. The hypergeometric superhyperbolic supercosecant via Jacobi–Luketype polynomials is defined as cosechn(α,β) (λz) =

1 (α,β) sinhn (λz)

,

(5.220)

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. In this case, we show that sinhn(α,β) (z)  1  (α,β) n = (λz) − n(α,β) (−λz) 2 1 (1 + α)n = 2 F1 (−n, n + α + β + 1; α + 1; λz) 2 n! 1 (1 + α)n − 2 F1 (−n, n + α + β + 1; α + 1; λz) 2 n! (1 + α)n = 2 Supersinh1 (−n, n + α + β + 1; α + 1; λz), n!

(5.221)

coshn(α,β) (λz)  1  (α,β) Rn = (λz) + Rn(α,β) (−λz) 2 1 (1 + α)n = 2 F1 (−n, n + α + β + 1; α + 1; λz) 2 n! 1 (1 + α)n − 2 F1 (−n, n + α + β + 1; α + 1; −λz) 2 n! (1 + α)n = 2 Supercosh1 (−n, n + α + β + 1; α + 1; λz), n!

(5.222)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

tanhn(α,β) (λz) (α,β)

=

sinhn

(λz)

(α,β) coshn (λz)

(5.223)

2 Supersinh1 (−n, n + α

+ β + 1; α + 1; λz) = Supercosh n + α + β + 1; α + 1; λz) (−n, 2 1 = 2 Supertanh1 (−n, n + α + β + 1; α + 1; λz), cotanhn(α,β) (λz) (α,β)

=

coshn

(λz)

(α,β) sinhn (λz)

(5.224)

2 Supercosh1 (−n, n + α

+ β + 1; α + 1; λz) = 2 Supersinh1 (−n, n + α + β + 1; α + 1; λz) = 2 Supercotanh1 (−n, n + α + β + 1; α + 1; λz), sechn(α,β) (λz) =

1 (α,β) coshn (λz)

(5.225)

,

and cosechn(α,β) (λz) = where i =

1 (α,β) sinhn (λz)

(5.226)

,

√ −1, Re (α) > −1, Re (β) > −1, −1 < z < 1, λ ∈ C, and n ∈ N0 . (α,β)

Definition 5.31. The hypergeometric superhyperbolic supersine sinhn (z), hyperge(α,β) ometric superhyperbolic supercosine coshn (z), hypergeometric superhyperbolic su(α,β) (α,β) pertangent tanhn (z), hypergeometric superhyperbolic supercotangent cotanhn (z), (α,β) hypergeometric superhyperbolic supersecant sechn (z), and hypergeometric super(α,β) are called the hypergeometric superhyperbolic hyperbolic supercosecant cosechn (z) functions via Jacobi–Luke-type polynomials. Due to the above definitions, we present the following theorems. Theorem 5.92. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  0

z

t c−1 (z − t)d−1 sinhn(α,β) (λt) dt =

(1 + α)n c+d−1 B (c, d) z n!

(5.227)

× 3 Supersinh2 (−n, n + α + β + 1, c; α + 1, c + d; λz). Proof. By means of the relations sinhn(α,β) (z) =

 1  (α,β) n (λz) − n(α,β) (−λz) 2

(5.228)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

and



z

t c−1 (z − t)d−1 2 F1 (−n, n + α + β + 1; β + 1; λt)dt

0

=z

c+d−1

333

(5.229)

B (c, d) 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; λz)

we have that  z t c−1 (z − t)d−1 sinhn(α,β) (λt) dt 0  (1 + α)n z c−1 t = (z − t)d−1 2 Supersinh1 (−n, n + α + β + 1; α + 1; λt)dt n! 0 (1 + α)n c+d−1 z = B (c, d) 3 Supersinh2 (−n, n + α + β + 1, c; +1, c + d; λz), n!

(5.230)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, c, d ∈ C\Z, and n ∈ N0 . As direct results, we present the following theorems. Theorem 5.93. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, and n ∈ N0 , then  z t c−1 (z − t)d−1 sinhn(α,β) (t) dt 0

=

(1 + α)n c+d−1 z B (c, d) 3 Supersinh2 (−n, n + α + β + 1, c; +1, c + d; z). n!

Theorem 5.94. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, λ ∈ R, and n ∈ N0 , then  1 (1 + α)n B (c, d) t c−1 (1 − t)d−1 sinhn(α,β) (t) dt = n! 0

(5.231)

(5.232)

× 3 Supersinh2 (−n, n + α + β + 1, c; α + 1, c + d; 1). Theorem 5.95. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then  1 (1 + α)n B (c, d) t c−1 (1 − t)d−1 sinhn(α,β) (λt) dt = n! (5.233) 0 × 3 Supersinh2 (−n, n + α + β + 1, c; α + 1, c + d; λ). Theorem 5.96. If Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 , then  z (1 + α)n c+d−1 z t c−1 (z − t)d−1  cosh(α,β) B (c, d) (λt) dt = n n! 0

(5.234)

× 3 Supercosh2 (−n, n + α + β + 1, c; α + 1, c + d; λz). Proof. By means of the relations  coshn(α,β) (z) =

 1  (α,β) n (λz) + n(α,β) (−λz) 2

(5.235)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and



z

t c−1 (z − t)d−1 2 F1 (−n, n + α + β + 1; β + 1; λt)dt

0

=z

c+d−1

(5.236)

B (c, d) 3 F2 (−n, n + α + β + 1, c; α + 1, c + d; λz)

we show that  z t c−1 (z − t)d−1 coshn(α,β) (λt) dt 0  (1 + α)n z c−1 t = (z − t)d−1 2 Supercosh1 (−n, n + α + β + 1; α + 1; λt)dt n! 0 (1 + α)n c+d−1 z = B (c, d) 3 Supercosh2 (−n, n + α + β + 1, c; +1, c + d; λz), n!

(5.237)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, c, d ∈ C\Z, and n ∈ N0 . As direct results, we give the following theorems. Theorem 5.97. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, and n ∈ N0 , then  z t c−1 (z − t)d−1  coshn(α,β) (t) dt 0

=

(1 + α)n c+d−1 z B (c, d) 3 Supercosh2 (−n, n + α + β + 1, c; +1, c + d; z). n!

Theorem 5.98. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, λ ∈ R, and n ∈ N0 , then  1 (1 + α)n t c−1 (1 − t)d−1 cosh(α,β) B (c, d) (t) dt = n n! 0

(5.238)

(5.239)

× 3 Supercosh2 (−n, n + α + β + 1, c; α + 1, c + d; 1). Theorem 5.99. If Re (α) > −1, Re (β) > −1, c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then  1 (1 + α)n B (c, d) t c−1 (1 − t)d−1 coshn(α,β) (λt) dt = n! (5.240) 0 × 3 Supercosh2 (−n, n + α + β + 1, c; α + 1, c + d; λ). Here we have n(α,β) (z) = coshn(α,β) (z) + sinh(α,β) (z) , n

(5.241)

where Re (α) > −1, Re (β) > −1, and −1 < z < 1. Using the above results, we get the following property. Property 5.5. We have sinn(α,β) (z) = Psinn(α,β) (−i − 2z) ,

(5.242)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

335

cos n(α,β) (z) = Pcos n(α,β) (−i − 2z) ,

(5.243)

tann(α,β) (z) = Ptann(α,β) (−i − 2z) ,

(5.244)

cotann(α,β) (z) = Pcotann(α,β) (−i − 2z) ,

(5.245)

sec n(α,β) (z) = Psec n(α,β) (−i − 2z) ,

(5.246)

cosec n(α,β) (z) = Pcosec n(α,β) (−i − 2z) ,

(5.247)

sinhn(α,β) (z) = Psinh(α,β) (−i − 2z) , n

(5.248)

coshn(α,β) (z) = Pcoshn(α,β) (−i − 2z) ,

(5.249)

tanhn(α,β) (z) = Pcoshn(α,β) (−i − 2z) ,

(5.250)

cotanhn(α,β) (z) = Pcoshn(α,β) (−i − 2z) ,

(5.251)

sechn(α,β) (z) = Psechn(α,β) (−i − 2z) ,

(5.252)

cosechn(α,β) (z) = Pcosechn(α,β) (−i − 2z) ,   z+i (α,β) − = Psinn(α,β) (z) , sinn 2   z+i = Pcos n(α,β) (z) , cos n(α,β) − 2   z+i (α,β) − = Ptann(α,β) (z) , tann 2   z+i = Pcotann(α,β) (z) , cotann(α,β) − 2   z+i (α,β) − = Psec n(α,β) (z) , sec n 2   z+i = Pcosec n(α,β) (z) , cosec n(α,β) − 2   z+i = Psinhn(α,β) (z) , sinhn(α,β) − 2   z+i (α,β) − = Pcoshn(α,β) (z) , coshn 2   z+i = Pcoshn(α,β) (z) , tanhn(α,β) − 2   z+i (α,β) − = Pcoshn(α,β) (z) , cotanhn 2   z+i = Psechn(α,β) (z) , sechn(α,β) − 2

(5.253) (5.254) (5.255) (5.256) (5.257) (5.258) (5.259) (5.260) (5.261) (5.262) (5.263) (5.264)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and

  z+i = Pcosechn(α,β) (z) , cosechn(α,β) − 2

(5.265)

where Re (α) > −1, Re (β) > −1, −1 < z < 1, and n ∈ N0 . √ Theorem 5.100. If α > −1, β > −1, i = −1, t ∈ C, −1 < z < 1, and k ∈ N0 , then we have the following generating functions: ∞ 

(α,β)

sink

(z) t k

k=0

 2α+β−1  −1 × ϑ1 (1 − t + ϑ1 )−α (1 + t + ϑ1 )−β − ϑ2−1 (1 − t + ϑ2 )−α (1 + t + ϑ2 )−β = i

(5.266)

and ∞ 

(α,β)

cos k

(z) t k

k=0

  = 2α+β−1 × ϑ1−1 (1 − t + ϑ1 )−α (1 + t + ϑ1 )−β + ϑ2−1 (1 − t + ϑ2 )−α (1 + t + ϑ2 )−β , (5.267)

where

and

1  2 ϑ1 = 1 − 2 (1 − 2iz) t + t 2

(5.268)

1  2 ϑ2 = 1 − 2 (1 + 2iz) t + t 2 ,

(5.269)

Proof. Using the relations ∞ 

(iz) t k = 2α+β ϑ1−1 (1 − t + ϑ1 )−α (1 + t + ϑ1 )−β

(5.270)

(−iz) t k = 2α+β ϑ2−1 (1 − t + ϑ2 )−α (1 + t + ϑ2 )−β ,

(5.271)

(α,β)

k

k=0

and ∞ 

(α,β)

k

k=0

where

and

1  2 ϑ1 = 1 − 2 (1 − 2iz) t + t 2

(5.272)

1  2 ϑ2 = 1 − 2 (1 + 2iz) t + t 2 ,

(5.273)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

337

we show that ∞ 

(α,β)

sink

(z) t k

k=0

=

∞  1   (α,β) (α,β) k (iz) − k (−iz) t k 2i

(5.274)

k=0

 2α+β−1  −1 = × ϑ1 (1 − t + ϑ1 )−α (1 + t + ϑ1 )−β − ϑ2−1 (1 − t + ϑ2 )−α (1 + t + ϑ2 )−β i and ∞ 

(α,β)

cos k

(z) t k

k=0

 1   (α,β) (α,β) k (iz) + k (−iz) t k 2 k=0   α+β−1 =2 × ϑ1−1 (1 − t + ϑ1 )−α (1 + t + ϑ1 )−β + ϑ2−1 (1 − t + ϑ2 )−α (1 + t + ϑ2 )−β . ∞

=

(5.275)

Theorem 5.101. If α > −1, β > −1, t ∈ C, −1 < z < 1, and k ∈ N0 , then we have the following generating functions: ∞ 

(α,β)

 sinhk

k=0

=2

α+β−1

×



(z) t k ϑ3−1 (1 − t

−α

+ ϑ3 )

−β

(1 + t + ϑ3 )

− ϑ4−1 (1 − t

+ ϑ4 )

−α

−β

(1 + t + ϑ4 )



(5.276)

and ∞ 

(α,β)

 coshk

(z) t k

k=0

  = 2α+β−1 × ϑ3−1 (1 − t + ϑ3 )−α (1 + t + ϑ3 )−β + ϑ4−1 (1 − t + ϑ4 )−α (1 + t + ϑ4 )−β , (5.277)

where

and

1  2 ϑ3 = 1 − 2 (1 − 2z) t + t 2

(5.278)

1  2 ϑ4 = 1 − 2 (1 + 2z) t + t 2 .

(5.279)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Proof. By means of the relations ∞ 

(z) t k = 2α+β ϑ3−1 (1 − t + ϑ3 )−α (1 + t + ϑ3 )−β

(5.280)

(−z) t k = 2α+β ϑ4−1 (1 − t + ϑ4 )−α (1 + t + ϑ4 )−β ,

(5.281)

(α,β)

k

k=0

and

∞ 

(α,β)

k

k=0

where

and

1  2 ϑ3 = 1 − 2 (1 − 2z) t + t 2

(5.282)

1  2 ϑ4 = 1 − 2 (1 + 2z) t + t 2 ,

(5.283)

we have that ∞ 

(α,β)

 sinhk

(z) t k

k=0 ∞  1   (α,β) (5.284) (α,β) k (z) − k (−z) t k 2 k=0   α+β−1 =2 × ϑ3−1 (1 − t + ϑ3 )−α (1 + t + ϑ3 )−β − ϑ4−1 (1 − t + ϑ4 )−α (1 + t + ϑ4 )−β

=

and ∞ 

(α,β)

 coshk

(z) t k

k=0 ∞  1   (α,β) (α,β) k = (z) + k (−z) t k 2 k=0   α+β−1 =2 × ϑ3−1 (1 − t + ϑ3 )−α (1 + t + ϑ3 )−β + ϑ4−1 (1 − t + ϑ4 )−α (1 + t + ϑ4 )−β .

(5.285)

Theorem 5.102 (Laplace transform). Let α, β, λ, z ∈ C, Re (α) > 0, Re (β) > 0, |z| < 1, and t ∈ [0, +∞). (α,β) Then the Laplace transform of the function t λ−1 sinhn (zt) is as follows:   L t λ−1 sinhn(α,β) (zt)   (1 + α)n  (λ) λ, −n, n + α + β + 1 z = ; , 3 Supersinh1 α+1 n! sλ s

(5.286)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

339

where the Laplace transform of a function f is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(5.287)

0

Theorem 5.103. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ ∈ N, then  z

1 t α−1 (z − t)β−1 sinhn(c,d) λt κ dt B (α, β) 0 (1 + c)n × zα+β−1 = n!   −n, n + c + d + 1, ακ , · · · , α+κ−1 ; κ κ λz , × 2+κ Supersinh1+κ α+β+κ−1 c + 1, α+β ; κ ,··· , κ

(5.288)

where λ is a constant. Theorem 5.104. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ, s ∈ N, then  0

z



t α−1 (z − t)β−1 sinhn(c,d) λt κ (z − t)s dt

(1 + c)n × B (α, β) × zα+β−1 n!   , βs , · · · , β+s−1 ; κ κ s s λzκ+s −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s × 2+κ+s Supersinh1+κ+s , α+β+κ+s−1 (κ + s)κ+s c + 1, α+β ; κ+s , · · · , κ+s (5.289) =

where λ is a constant. Theorem 5.105. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ ∈ N, then  0

z



t α−1 (z − t)β−1 sinhn(c,d) λ (z − t)s dt

(1 + c)n × B (α, β) × zα+β−1 n!   −n, n + c + d + 1, βs , · · · , β+s−1 ; s s × 2+s Supersinh1+s λz , α+β+s−1 c + 1, α+β ; s ,··· , s =

where λ is a constant.

(5.290)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 5.106. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

1 t α−1 (1 − t)β−1 sinhn(c,d) λt κ dt B (α, β) 0 (1 + c)n = n!   ; −n, n + c + d + 1, ακ , · · · , α+κ−1 κ λ , × 2+κ Supersinh1+κ α+β+κ−1 ; c + 1, α+β κ ,··· , κ

(5.291)

where λ is a constant. Theorem 5.107. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then  0

=

1



t α−1 (1 − t)β−1 sinhn(c,d) λt κ (1 − t)s dt

(1 + c)n × B (α, β) n!

× 2+κ+s Supersinh1+κ+s

 , βs , · · · , β+s−1 ; −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s α+β+κ+s−1 ; c + 1, α+β κ+s , · · · , κ+s

 κ κ ss λ , (κ + s)κ+s (5.292)

where λ is a constant. Theorem 5.108. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  0

1



t α−1 (1 − t)β−1 sinhn(c,d) λ (1 − t)s dt

(1 + c)n × B (α, β) n!   −n, n + c + d + 1, βs , · · · , β+s−1 ; s × 2+s Supersinh1+s λ , α+β+s−1 c + 1, α+β ; s ,··· , s =

(5.293)

where λ is a constant. Theorem 5.109. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

1 λ (t − 1)κ dt (t − 1)α−1 (1 + t)β−1 sinh(c,d) n B (α, β) −1 (1 + c)n × 2α+β−1 = n!   ; −n, n + c + d + 1, ακ , · · · , α+κ−1 κ λ2κ , × 2+κ Supersinh1+κ α+β+κ−1 c + 1, α+β , · · · , ; κ κ where λ is a constant.

(5.294)

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

341

Theorem 5.110. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then  1

(t − 1)α−1 (1 + t)β−1 sinhn(c,d) λ (t − 1)κ (1 + t)s dt −1

(1 + c)n × B (α, β) × 2α+β−1 n!   , βs , · · · , β+s−1 ; κ κ s s λ2κ+s −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s × 2+κ+s Supersinh1+κ+s , α+β+κ+s−1 (κ + s)κ+s c + 1, α+β ; κ+s , · · · , κ+s (5.295) =

where λ is a constant. Theorem 5.111. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

λ (1 + t)s dt (t − 1)α−1 (1 + t)β−1 sinh(c,d) n −1

(1 + c)n × B (α, β) × 2α+β−1 n!   −n, n + c + d + 1, βs , · · · , β+s−1 ; s s × 2+s Supersinh1+s λ2 , α+β+s−1 c + 1, α+β ; s ,··· , s =

(5.296)

where λ is a constant. Theorem 5.112 (Laplace transform). Let α, β, λ, z ∈ C, Re (α) > 0, Re (β) > 0, |z| < 1, and t ∈ [0, +∞). (α,β) Then the Laplace transform of the function t λ−1 cos n (zt) is as follows:   L t λ−1 coshn(α,β) (zt)   (1 + α)n  (λ) λ, −n, n + α + β + 1 z = , ; 3 Supercosh1 α+1 n! sλ s where the Laplace transform of a function f is



L [f (t)] = f (s) =

∞

e−st f (t) dt.

(5.297)

(5.298)

0

Theorem 5.113. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ ∈ N, then  z

1 t α−1 (z − t)β−1 coshn(c,d) λt κ dt B (α, β) 0 (1 + c)n = × zα+β−1 (5.299) n!   α+κ−1 −n, n + c + d + 1, ακ , α+1 ; κ κ ,··· , κ λz , × 2+κ Supercosh1+κ α+β α+β+1 α+β+κ−1 c + 1, κ , κ , · · · , ; κ where λ is a constant.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 5.114. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ, s ∈ N, then  z κ

λt (z − t)s dt t α−1 (z − t)β−1 cosh(c,d) n 0

(1 + c)n × B (α, β) × zα+β−1 n!   , βs , · · · , β+s−1 ; κ κ s s λzκ+s −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s × 2+κ+s Supercosh1+κ+s , α+β+κ+s−1 (κ + s)κ+s ; c + 1, α+β κ+s , · · · , κ+s (5.300) =

where λ is a constant. Theorem 5.115. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, |z| < 1, and κ ∈ N, then  z

t α−1 (z − t)β−1 coshn(c,d) λ (z − t)s dt 0

(1 + c)n × B (α, β) × zα+β−1 n!   β+s−1 −n, n + c + d + 1, βs , β+1 ; s s ,··· , s λz , × 2+s Supercosh1+s α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s =

(5.301)

where λ is a constant. Theorem 5.116. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

1 t α−1 (1 − t)β−1 coshn(c,d) λt κ dt B (α, β) 0 (1 + c)n = n!   α+κ−1 ; −n, n + c + d + 1, ακ , α+1 κ ,··· , κ λ , × 2+κ Supercosh1+κ α+β+1 c + 1, α+β , · · · , α+β+κ−1 ; κ , κ κ

(5.302)

where λ is a constant. Theorem 5.117. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then  1

t α−1 (1 − t)β−1 coshn(c,d) λt κ (1 − t)s dt 0

=

(1 + c)n × B (α, β) n!

× 2+κ+s Supercosh1+κ+s



, βs , · · · , β+s−1 ; −n, n + c + d + 1, ακ , · · · , α+κ−1 κ s α+β+κ+s−1 ; c + 1, α+β κ+s , · · · , κ+s

 κ κ ss λ , (κ + s)κ+s (5.303)

where λ is a constant.

Chapter 5 • Hypergeometric supertrigonometric and superhyperbolic functions

343

Theorem 5.118. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

t α−1 (1 − t)β−1 coshn(c,d) λ (1 − t)s dt 0

(1 + c)n × B (α, β) n!   −n, n + c + d + 1, βs , · · · , β+s−1 ; s × 2+s Supercosh1+s λ , α+β+s−1 ; c + 1, α+β s ,··· , s =

(5.304)

where λ is a constant. Theorem 5.119. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

1 (t − 1)α−1 (1 + t)β−1 coshn(c,d) λ (t − 1)κ dt B (α, β) −1 (1 + c)n = × 2α+β−1 n!   ; κ −n, n + c + d + 1, ακ , · · · , α+κ−1 κ λ2 , × 2+κ Supercosh1+κ α+β+κ−1 ; c + 1, α+β κ ,··· , κ

(5.305)

where λ is a constant. Theorem 5.120. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ, s ∈ N, then  1

(t − 1)α−1 (1 + t)β−1 coshn(c,d) λ (t − 1)κ (1 + t)s dt −1

(1 + c)n × B (α, β) × 2α+β−1 n!   β β+s−1 κ s s λ2κ+s , , · · · , ; −n, n + c + d + 1, ακ , · · · , α+κ−1 κ κ s s × 2+κ+s Supercosh1+κ+s , α+β+κ+s−1 (κ + s)κ+s c + 1, α+β , · · · , ; κ+s κ+s (5.306) =

where λ is a constant. Theorem 5.121. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (d) > 0, and κ ∈ N, then  1

(t − 1)α−1 (1 + t)β−1 coshn(c,d) λ (1 + t)s dt −1

(1 + c)n × B (α, β) × 2α+β−1 n!   −n, n + c + d + 1, βs , · · · , β+s−1 ; s λ2s , × 2+s Supercosh1+s α+β+s−1 c + 1, α+β , · · · , ; s s =

where λ is a constant. For more detail on the Jacobi–Luke polynomials, see [14].

(5.307)

6 Hypergeometric supertrigonometric and superhyperbolic functions via Laguerre polynomials 6.1 Laguerre polynomials In 1878, Laguerre proposed the well-known Laguerre polynomial series, which is called the Laguerre polynomials [180]. In this section, we introduce the definition, properties, and theorems for the Laguerre polynomials. We consider the Laplace transform for the Laguerre polynomials in detail.

6.1.1 Definition, properties, and theorems for the Laguerre polynomials Definition 6.1. [Laguerre (1878)] (α) The Laguerre polynomial of degree n, denoted as Ln (z), is defined by [180] L(α) n (z) =

n   (α + n + 1) (−z)k ,  (α + k + 1) k! (n − k)

(6.1)

k=0

where n ∈ N0 . The result was discovered by Laguerre in 1878 [180]. Property 6.1. [Rodrigues (1816); Laguerre (1878)] The Laguerre polynomial of degree n can be written as L(α) n (z) =

ez z−α d n  −z n+α  e z , n! dzn

(6.2)

where n ∈ N, α > −1, and z ∈ C. The form of the presented result was proposed by Rodrigues in 1816 [160], and its generalized version was discovered by Laguerre in 1878 [180]. The detailed report was discussed by in Erdélyi et al. in 1953 [88]. From Eq. (6.1) we have (α)

L0 (z) = 1,

(6.3)

L(α) 1 (z) = 1 + α − z,

(6.4)

An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions https://doi.org/10.1016/B978-0-12-824154-7.00012-X Copyright © 2021 Elsevier Inc. All rights reserved.

345

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

1 1 (1 + α) (2 + α) − (2 + α) z + z2 , 2 2

(6.5)

1 1 1 (1 + α) (2 + α) (3 + α) − (2 + α) (3 + α) z2 − z3 . 6 2 6

(6.6)

L(α) 2 (z) = and (α)

L3 (z) =

Property 6.2. The Laguerre polynomial of degree n can be written as [88] L(α) n (z) =

(α + 1)n 1 F1 (−n; α + 1; z), n!

(6.7)

where n ∈ N, α > −1, and z ∈ C. Property 6.3. The Laguerre polynomial [25] L(α) n (z) (α + 1)n = 1 F1 (−n; α + 1; z) n! n   (α + n + 1) (−z)k =  (α + k + 1) k! (n − k) =

k=0 ez z−α

n!

(6.8)

d n  −z n+α  e z dzn

is a solution of the Laguerre differential equation z

d 2ϕ dϕ − nϕ = 0, + (α − z + 1) dz dz2

(6.9)

where z ∈ C, n ∈ N, α > −1, and |z| < 1. Property 6.4. We have [181] (1 + t)α e−zt =

∞ 

Ln(α−n) (z) t n ,

(6.10)

n=0

where z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Property 6.5. We have [181] 2  √ z 1− 1−4t

∞

 4t 2α e − Ln(α+n) (z) t n ,  = √ √ 1 − 4t 1 + 1 − 4t n=0 where z ∈ C, n ∈ N, α > −1, and |z| < 1.

(6.11)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

Property 6.6 (Brafman theorem). [Brafman (1957)] We have [182]  e−z zn+α ez z−α L(α) dx, = (z) n 2πi  (x − z)n+1

347

(6.12)

where  is the Brafman path. Theorem 6.1. Let z ∈ C, n ∈ N, α > −1, and |z| < 1. Then [25] d n (α) L (z) = (−1)n . dzn n

(6.13)

Theorem 6.2. The orthogonal relation for the Laguerre polynomials is given as follows [25]:  ∞  (α + 1 + n) −z α (α) δmn , L(α) (6.14) n (z) Lm (z) e z dz = n! 0 where α > −1 and n, m ∈ N. Theorem 6.3 (Hille theorem). [Hille (1926)] If α > −1 and n ∈ N, then f (z) =

∞ 

Wn L(α) n (z),

(6.15)

n=0

where



∞ 0

−z L(α) n (z) e f (z) dz = Wn .

(6.16)

The result is the Fourier–Laguerre series proposed by Hille in 1926 [183]. In this case we have [25]    ∞   − 12 1 2ez 2 (6.17) e−t t 2n cos 2 (zt) 2 dt Ln (z) = √ n! π 0 and

 

Ln

1 2

(z) =

2ez √ n! πz



∞

  1 2 e−t t 2n cos 2 (zt) 2 dt.

(6.18)

0

Moreover, we have [25] e−z L(α) n (z) =

1  (β − α)

 z

∞

(t − z)β−α−1 e−t L(β) n (t)dt,

d (α) (α+1) L (z) = −Ln−1 (z) , dz n d  α (α)  z Ln (z) = (n + α) zα−1 Ln(α−1) (z) , dz

(6.19) (6.20) (6.21)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and

d  −z (α)  e Ln (z) = −e−z L(α+1) (z) , n dz

(6.22)

d  α −z (α)  (α−1) z e Ln (z) = (n + 1) zα−1 e−z Ln+1 (z) . dz

(6.23)

Theorem 6.4. We have L(α) n (z) =

1  (β − α)



∞

z

(t − z)β−α−1 e−(t−z) L(β) n (t)dt.

(6.24)

  (t − z)β−α−1 e−(t−z) L(β) n (t) dt,

(6.25)

Theorem 6.5. We have   L(α) n (z) =

1  (β − α)

 z

∞

  (α) (α) where Ln (z) = ( ◦ L) (z) with L (z) = Ln (z). Theorem 6.6. If α > β, then [25] z

α

L(α) n (z) =

 (n + α + 1)  (α − β)  (n + β + 1)



z

0

(z − t)α−β−1 t β L(β) n (t) dt,

(6.26)

where α > −1. Theorem 6.7. We have [25] L(α+β+1) (z + t) = n

n 

(β)

L(α) k (z) Ln−k (t),

(6.27)

k=0

where α > −1. Theorem 6.8. We have [25,159] L(α+β+1) (λz + t) = n

n 

(α)

(β)

Lk (λz) Ln−k (t),

(6.28)

k=0

L(α+β+1) (z + λt) = n

n 

(α)

(β)

Lk (z) Ln−k (λt),

(6.29)

k=0

and L(α+β+1) (λz + γ t) = n

n  k=0

where λ, γ ∈ C and α > −1.

(α)

(β)

Lk (λz) Ln−k (γ t),

(6.30)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

349

Theorem 6.9. We have [25] zα e−z L(α) n (z) =

∞  k=n

 (n + k + β − α)  (k + α − 1)  (k + 1) (β) L (z) zβ e−z ,  (β − α)  (n − k + 1)  (k + 1)  (n + β + 1) k

(6.31)

where α > (β − 1) /2. Theorem 6.10 (Feldheim theorem). [Feldheim (1943)] We have [184] (α)

Ln+m (z) (α) Ln+m (0)

=



 (α + 1)  (β + 1)  (α − β)

1

(β)

(1 − t)α−β−1 t β

(α−β−1)

Lm (zt) Ln

0

(β) Lm (0)

(z (1 − t))

(α−β−1) Ln (0)

dt,

(6.32)

where α > −1. Theorem 6.11. We have [25] n 

L(α) n (z) =

k=0

 (n − k + α − β) (β) L (z),  (n − k + 1)  (α − β) k

(6.33)

where α > −1. Property 6.7. [Brafman (1957)] We have [182]  ∞  n n 1 b b 1 1 α 1 α k + , 1 + ; v 2 L(α) 4 F3 − , − + , , + ; , k (z) t 2 2 2 2 2 2 2 2 2 2 k=0  −zvt 1 −zt = e 1−t (1 − t)b−α−1 (1 − t − vt)b 1 F1 b; α + 1; 2 (1 − t) (1 − t − vt)  zvt 1 −zt . + e 1−t (1 − t)b−α−1 (1 − t + vt)b 1 F1 b; α + 1; 2 (1 − t) (1 − t + vt)

(6.34)

Theorem 6.12 (Weisner theorem). [Weisner (1955)] We have [185] ∞ 

2 F1 (−n, b; α

(α)

+ 1; v) Lk (z) t k

k=0

 zvt 1 −zt . = e 1−t (1 − t)−1−α+b (1 − t + vt)−b 1 F1 b; α + 1; 2 (1 − t) (1 − t + vt)

(6.35)

Theorem 6.13. We have [45]  0

where α > −1, n, m ∈ N, and m < n.

∞

m e−z zα L(α) n (z) z dz = 0,

(6.36)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.14. We have [45] n  k=0

(α) (α) (α) (α) k! (n + 1)! Ln (z) Ln+1 (t) − Ln+1 (z) Ln (t) (α) (α) , Lk (z) Lk (t) = z−t (1 + α)k (1 + α)k

(6.37)

where α > −1. Theorem 6.15. We have L(α) ϕ (1) = −

sin (ϕπ)  (α + 1 + ϕ) π  (α + 1)



∞

e−t t −ϕ−1 0 F1 (−; α + 1; t)dt,

(6.38)

e−t t −ϕ−1 0 F1 (−; α + 1; t)dt,

(6.39)

0

where ϕ ∈ C\Z and α > −1. Proof. We have L(α) ϕ (1) = −

sin (ϕπ)  (α + 1 + ϕ) π  (α + 1)



∞

0

since 1 F1 (a; c; 1) =

1  (a)



∞

e−t t a−1 0 F1 (−; c; t)dt,

(6.40)

0

1 sin (ϕπ) =− ,  (−ϕ)  (1 + ϕ) π

(6.41)

and (α + 1)ϕ =

 (α + ϕ + 1)  (α + 1)

(6.42)

where ϕ ∈ C\Z and α > −1. Theorem 6.16. We have (α)  (z) − 12

L

  1  ∞ 1 1  α+ 2 = e−t t − 2 0 F1 (−; α + 1; zt)dt π  (α + 1) 0   ∞ 1 1 1 e−t t − 2 0 F1 (−; α + 1; zt)dt, = − π 2 α+1 0

(6.43)

where α > −1, and 0 F1 (−; α; z) =

∞  1 zn . (α)n n! n=0

(6.44)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

351

Theorem 6.17. We have

  ∞  n+ 1 ∞   2 zn zn 1 1 1  L(0) 1  (z) = √ − =√ . −2  (n + 2) n 2 n+2 n π π n=0

(6.45)

n=0

Proof. We have (0)  (z) − 12

L

1 =√ π 1 =√ π 1 =√ π 1 =√ π



 1 (zt)n e t dt  (2 + n) n! 0 n=0  ∞ ∞  1 zn e−t t n− 2 dt  (n + 2)  (1 + n) 0 n=0   ∞  n+ 1  2 zn  (n + 2) n n=0 ∞   1 zn − , 2 n+2 n 

∞

−t − 12

∞ 

(6.46)

n=0

since 0 F1 (−; 1; z) =

∞ ∞  zn 1 zn  1 =  (2 + n) n! (1)n n! n=0

and

(6.47)

n=0



∞

 (n) =

e−t t n−1 dt.

(6.48)

0

Theorem 6.18. We have (0)  (0) = 1. − 12

L

(6.49)

Proof. Taking z = 0, we have   1  2 1 (0)   = 1, L 1 (0) = √ −2 π  (2) where 

 √ 1 = π. 2

(6.50)

(6.51)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.19. We have L(α) ϕ (z) = −

sin (ϕπ) π



1

t −ϕ−1 (1 − t)α+ϕ ezt dt,

(6.52)

0

where ϕ ∈ C\Z and α > −1. Proof. We have the equation of the form L(α) ϕ (z) =

(α + 1)ϕ 1 F1 (−ϕ; α + 1; z), ϕ!

(6.53)

which leads to the expression L(α) n (z) (α + 1)ϕ = 1 F1 (−ϕ; α + 1; z) ϕ!  1 (α + 1)ϕ  (α + 1) = t ϕ−1 (1 − t)α+ϕ ezt dt  (1 + ϕ)  (−ϕ)  (α + 1 + ϕ) 0  1 (α + 1)ϕ 1 t ϕ−1 (1 − t)α+ϕ ezt dt =  (1 + ϕ)  (−ϕ) (α + 1)ϕ 0  1 1 = t ϕ−1 (1 − t)α+ϕ ezt dt  (1 + ϕ)  (−ϕ) 0  1 1 t ϕ−1 (1 − t)α+ϕ ezt dt =  (1 + ϕ)  (−ϕ) 0  sin (ϕπ ) 1 −ϕ−1 t =− (1 − t)α+ϕ ezt dt π 0 since 1 F1 (−ϕ; α

+ 1; z) =

 (α + 1)  (−ϕ)  (α + 1 + ϕ)



1

t ϕ−1 (1 − t)α−ϕ ezt dt.

(6.54)

(6.55)

0

6.1.2 Some integral representations via Laguerre polynomials We now present some integral representations via Laguerre polynomials. Theorem 6.20. If ϕ ∈ C\Z and α > −1, then we have  ∞  1 1 t −ϕ−1 (1 − t)α+ϕ ezt dt = e−t t −ϕ−1 0 F1 (−; α + 1; zt)dt. (α + 1)ϕ 0 0 Proof. We have the integral representations  sin (ϕπ) 1 −ϕ−1 L(α) t = − (z) (1 − t)α+ϕ ezt dt ϕ π 0

(6.56)

(6.57)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

353

and L(α) ϕ (z) = −

sin (ϕπ)  (α + 1 + ϕ) π  (α + 1)



∞

e−t t −ϕ−1 0 F1 (−; α + 1; zt)dt

(6.58)

0

such that  sin (ϕπ ) 1 −ϕ−1 t (1 − t)α+ϕ ezt dt π 0  sin (ϕπ )  (α + 1 + ϕ) ∞ −t −ϕ−1 e t =− 0 F1 (−; α + 1; zt)dt, π  (α + 1) 0

−

(6.59)

which leads to  1 1 t −ϕ−1 (1 − t)α+ϕ ezt dt (α + 1)ϕ 0  ∞ = e−t t −ϕ−1 0 F1 (−; α + 1; zt)dt 0  ∞ e−t t −ϕ−1 0 F1 (−; α + 1; zt)dt. =

(6.60)

0

Theorem 6.21 (Laplace transform). Let α, z, λ ∈ C, n ∈ N0 , and t ∈ [0, +∞). (α) Then the Laplace transform of t λ−1 Ln (zt) is as follows:

L t λ−1 L(α) n (zt)   (α + 1)n λ−1 t =L 1 F1 (−n; α + 1; zt) n!   ∞ −st (α + 1)n λ−1 = t e 1 F1 (−n; α + 1; zt) dt n! 0  (α + 1)n  (λ) λ, −n z = · λ · 2 F1 , ; α+1 s n! s

(6.61)

where the Laplace transform of a function f is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

0

At the moment, it is not difficult to derive the following theorems:

(6.62)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.22. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and κ ∈ N, then 

 κ (c + 1)n α+β−1 t α−1 (z − t)β−1 L(c) ·z n λt dt = n! 0

 ; −n, ακ , · · · , α+κ−1 κ κ × 1+κ F1+κ , α+β α+β+κ−1 λz ; c + 1, κ , · · · , κ

1 B (α, β)

z

(6.63)

where λ is a constant. Theorem 6.23. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and κ, s ∈ N, then we have  z  κ  1 s t α−1 (z − t)β−1 L(c) n λt (z − t) dt B (α, β) 0 (c + 1)n α+β−1 = ·z n!

 β β+s−1 κ s s λzκ+s , , · · · , ; −n, ακ , · · · , α+κ−1 κ κ s s × 1+κ+s F1+κ+s κ+s , α+β+κ+s−1 c + 1, α+β , · · · , ; (κ + s) κ+s κ+s

(6.64)

where λ is a constant. Theorem 6.24. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and s ∈ N, then we have  z   1 s t α−1 (z − t)β−1 L(c) n λ (z − t) dt B (α, β) 0 (c + 1)n α+β−1 ·z = n!

 β+s−1 −n, βs , β+1 ; s ,··· , s s × 1+s F1+s λz , α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s

(6.65)

where λ is a constant. Theorem 6.25. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then 

 κ t α−1 (1 − t)β−1 L(c) n λt dt 0 

−n, ακ , · · · , α+κ−1 ; (c + 1)n κ = × 1+κ F1+κ α+β+κ−1 λ , n! ; c + 1, α+β κ ,··· , κ

1 B (α, β)

where λ is a constant.

1

(6.66)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

355

Theorem 6.26. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  1  κ  1 s t α−1 (1 − t)β−1 L(c) n λt (1 − t) dt B (α, β) 0 (c + 1)n = (6.67) n!

 , βs , · · · , β+s−1 ; κ κ ss λ −n, ακ , · · · , α+κ−1 κ s × 1+κ+s F1+κ+s κ+s , α+β+κ+s−1 ; (κ + s) c + 1, α+β κ+s , · · · , κ+s where λ is a constant. Theorem 6.27. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 s t α−1 (1 − t)β−1 L(c) n λ (1 − t) dt B (α, β) 0 (c + 1)n = n!

 β+s−1 −n, βs , β+1 ; s ,··· , s λ , × 1+s F1+s α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s

(6.68)

where λ is a constant. Theorem 6.28. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then 

  (c + 1)n α+β−1 κ ·2 (1 − t)α−1 (1 + t)β−1 L(c) n λ (1 − t) dt = n! −1

 −n, ακ , · · · , α+κ−1 ; κ κ × 1+κ F1+κ , α+β α+β+κ−1 λ2 ; c + 1, κ , · · · , κ

1 B (α, β)

1

(6.69)

where λ is a constant. Theorem 6.29. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  1   1 κ s (1 − t)α−1 (1 + t)β−1 L(c) n λ (1 − t) (1 + t) dt B (α, β) −1 (c + 1)n α+β−1 ·2 = (6.70) n!

 , βs , · · · , β+s−1 ; κ κ s s λ2κ+s −n, ακ , · · · , α+κ−1 κ s × 1+κ+s F1+κ+s κ+s , α+β α+β+κ+s−1 ; (κ + s) c + 1, κ+s , · · · , κ+s where λ is a constant.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.30. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 s (1 − t)α−1 (1 + t)β−1 L(c) n λ (1 + t) dt B (α, β) −1 (c + 1)n α+β−1 = ·2 n!

 β+s−1 −n, βs , β+1 , · · · , ; s s × 1+s F1+s λ2s , α+β+1 α+β+s−1 c + 1, α+β , , · · · , ; s s s

(6.71)

where λ is a constant.

6.1.3 The hypergeometric supertrigonometric functions via Laguerre polynomials In this section, we propose the hypergeometric supertrigonometric functions via Laguerre polynomials. Definition 6.2. The hypergeometric supersine via Laguerre polynomials is defined as + 1; z)  1  (α) Ln (iz) − L(α) = (−iz) n 2i n   (α + n + 1) (−1)k z2κ+1 = ,  (α + 2κ + 2) (2κ + 1)! (n − (2κ + 1))

1 Lsupersin1 (−n; α

(6.72)

k=0

where z ∈ C, n ∈ N, α > −1, and |z| < 1. Definition 6.3. The hypergeometric supercosine via Laguerre polynomials is defined as + 1; z)   1 (α) Ln (iz) + L(α) = n (−iz) 2 n   (α + n + 1) (−1)k z2k = ,  (α + 2κ + 1) (2κ)! (n − 2κ)

1 Lsupercos 1 (−n; α

(6.73)

k=0

where z ∈ C, n ∈ N, α > −1, and |z| < 1. Definition 6.4. The hypergeometric supertangent via Laguerre polynomials is defined as 1 Lsupertan1 (−n; α

+ 1; z) =

where z ∈ C, n ∈ N, α > −1, and |z| < 1.

+ 1; z) , 1 Lsupercos 1 (−n; α + 1; z) 1 Lsupersin1 (−n; α

(6.74)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

357

Definition 6.5. The hypergeometric supercotangent via Laguerre polynomials is defined as 1 Lsupercot 1 (−n; α

+ 1; z) =

+ 1; z) , Lsupersin α + 1; z) (−n; 1 1

1 Lsupercos 1 (−n; α

(6.75)

where z ∈ C, n ∈ N, α > −1, and |z| < 1. Definition 6.6. The hypergeometric supersecant via Laguerre polynomials is defined as 1 Lsupersec 1 (−n; α

+ 1; z) =

1 , 1 Lsupercos 1 (−n; α + 1; z)

(6.76)

where z ∈ C, n ∈ N, α > −1, and |z| < 1. Definition 6.7. The hypergeometric supercosecant via Laguerre polynomials is defined as 1 Lsupercosec 1 (−n; α

+ 1; z) =

1 , 1 Lsupersin1 (−n; α + 1; z)

(6.77)

where z ∈ C, n ∈ N, α > −1, and |z| < 1. Definition 6.8. The hypergeometric supersine 1 Lsupersin1 (−n; α + 1; z), hypergeometric supercosine 1 Lsupercos 1 (−n; α + 1; z), hypergeometric supertangent 1 Lsupertan1 (−n; α + 1; z), hypergeometric supercotangent 1 Lsupercot 1 (−n; α + 1; z), hypergeometric supersecant 1 Lsupersec 1 (−n; α + 1; z), and hypergeometric supercosecant 1 Lsupercosec 1 (−n; α + 1; z) are called the hypergeometric supertrigonometric functions via Laguerre polynomials. In this case, we derive the following property. Property 6.8. Let z ∈ C, n ∈ N, α > −1, and |z| < 1. Then we have + 1; −z) = −1 Lsupersin1 (−n; α + 1; z),

(6.78)

+ 1; −z) = 1 Lsupercos 1 (−n; α + 1; z),

(6.79)

1 Lsupertan1 (−n; α

+ 1; −z) = −1 Lsupertan1 (−n; α + 1; z),

(6.80)

1 Lsupercot 1 (−n; α

+ 1; −z) = −1 Lsupercot 1 (−n; α + 1; z),

(6.81)

+ 1; −z) = 1 Lsupersec 1 (−n; α + 1; z) ,

(6.82)

+ 1; −z) = −1 Lsupercosec 1 (−n; α + 1; z).

(6.83)

1 Lsupersin1 (−n; α

1 Lsupercos 1 (−n; α

1 Lsupersec 1 (−n; α

and 1 Lsupercosec 1 (−n; α

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Clearly, we show that L(α) n (iλz) =

n   (α + n + 1) (−iλz)k  (α + k + 1) k! (n − k) k=0

n   (α + n + 1) (−1)k (λz)2k =  (α + 2κ + 1) (2κ)! (n − 2κ) k=0 n 

+i

k=0

(6.84)

 (α + n + 1) (−1)k (λz)2κ+1  (α + 2κ + 2) (2κ + 1)! (n − (2κ + 1))

= 1 Lsupercos 1 (−n; α + 1; λz) + i1 Lsupersin1 (−n; α + 1; λz) and L(α) n (−iλz) n   (α + n + 1) (iλz)k =  (α + k + 1) k! (n − k) k=0

n   (α + n + 1) (−1)k (λz)2k =  (α + 2κ + 1) (2κ)! (n − 2κ) k=0 n 

−i

k=0

(6.85)

 (α + n + 1) (−1)k (λz)2κ+1  (α + 2κ + 2) (2κ + 1)! (n − (2κ + 1))

= 1 Lsupercos 1 (−n; α + 1; λz) − i1 Lsupersin1 (−n; α + 1; λz) , where z ∈ C, k, n ∈ N, α > −1, |z| < 1, and i =

√ −1.

As a direct result, we have the following property. Property 6.9. Let z ∈ C, k, n ∈ N, α > −1, |z| < 1, and i =

√ −1. Then we have

L(α) n (iz) = 1 Lsupercos 1 (−n; α + 1; z) + i1 Lsupersin1 (−n; α + 1; z)

(6.86)

L(α) n (−iz) = 1 Lsupercos 1 (−n; α + 1; z) − i1 Lsupersin1 (−n; α + 1; z) .

(6.87)

and

Generally, we have the following result: √ If z ∈ C, k, n ∈ N, α > −1, |z| < 1, and i = −1, then we have L(α) n (iλz) = 1 Lsupercos 1 (−n; α + 1; λz) + i1 Lsupersin1 (−n; α + 1; λz)

(6.88)

L(α) n (−iλz) = 1 Lsupercos 1 (−n; α + 1; λz) − i1 Lsupersin1 (−n; α + 1; λz) .

(6.89)

and

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

359

We now present the series of the hypergeometric supertrigonometric functions via Laguerre polynomials. Series 1 Now we structure the series

(z) =

∞ 

  φ (κ) L(α) , (iκz) n

(6.90)

κ=0

which can be represented in the form

(z) = γ +

∞       ϕ (κ) 1 Lsupercos 1 (−n; α + 1; κz) + ψ (κ) 1 Lsupersin1 (−n; α + 1; κz) ,

(6.91)

κ=1

where ϕ (0) − iψ (0) , 2 ϕ (κ) + iψ (κ) φ (κ) = , 2 γ=

(6.92) (6.93)

and ϕ (κ) − iψ (κ) (6.94) 2 are the coefficients of the hypergeometric supertrigonometric functions via Laguerre polynomials. φ (−κ) =

Series 2 If (−z) = − (z), then there exists the series of the form

(z) =

∞     ψ (κ) 1 Lsupersin1 (−n; α + 1; κz) ,

(6.95)

κ=1

where ψ (κ) are the coefficients of the hypergeometric supertrigonometric functions via Laguerre polynomials. Series 3 If (−z) = (z), then there exists the series of the form

(z) = γ +

∞     ϕ (κ) 1 Lsupercos 1 (−n; α + 1; κz) ,

(6.96)

κ=1

where γ and ϕ (κ) are the coefficients of the hypergeometric supertrigonometric functions via Laguerre polynomials.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

In this case, we have the followings result: By means of the representations L(α) n (iλz) =

n   (α + n + 1) (−iλz)k  (α + k + 1) k! (n − k) k=0

n   (α + n + 1) (−1)k (λz)2k =  (α + 2κ + 1) (2κ)! (n − 2κ) k=0 n 

+i

k=0

(6.97)

 (α + n + 1) (−1)k (λz)2κ+1  (α + 2κ + 2) (2κ + 1)! (n − (2κ + 1))

and L(α) n (−iλz) = =

n   (α + n + 1) (iλz)k  (α + k + 1) k! (n − k) k=0 n 

 (α + n + 1) (λz)2k  (α + 2κ + 1) (2κ)! (n − 2κ)

k=0 n 

−i

k=0

(6.98)

 (α + n + 1) (λz)2κ+1  (α + 2κ + 2) (2κ + 1)! (n − (2κ + 1))

we have that + 1; λz)  1  (α) Ln (iλz) − L(α) = (−iλz) n 2i n   (α + n + 1) (−1)k (λz)2κ+1 , =  (α + 2κ + 2) (2κ + 1)! (n − (2κ + 1))

(6.99)

+ 1; λz)   1 (α) Ln (iλz) + L(α) = n (−iλz) 2 n   (α + n + 1) (λz)2k = ,  (α + 2κ + 1) (2κ)! (n − 2κ)

(6.100)

1 Lsupersin1 (−n; α

k=0

1 Lsupercos 1 (−n; α

k=0

1 Lsupertan1 (−n; α

+ 1; λz) =

1 Lsupercot 1 (−n; α

+ 1; λz) =

+ 1; λz) , Lsupercos α + 1; λz) (−n; 1 1

(6.101)

+ 1; λz) , 1 Lsupersin1 (−n; α + 1; λz)

(6.102)

1 Lsupersin1 (−n; α

1 Lsupercos 1 (−n; α

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

1 Lsupersec 1 (−n; α

+ 1; λz) =

1 , 1 Lsupercos 1 (−n; α + 1; λz)

361

(6.103)

and 1 Lsupercosec 1 (a; c; λz) =

1 , 1 Lsupersin1 (−n; α + 1; λz)

(6.104)

√ where z ∈ C, n ∈ N, α > −1, |z| < 1, and i = −1. We now consider the theorems for the integral representations for the hypergeometric supertrigonometric functions via Laguerre polynomials. √ Theorem 6.31. If z, λ ∈ C, n ∈ N, α > −1, |z| < 1, and i = −1, then we have  1  (α) Ln (iλz) − L(α) (−iλz) n 2i (α + 1)n = 1 Supersin1 (−n; α + 1; λz) n! ∞ (α + 1)n  (−n)k (−1)k (λz)2k+1 = n! (a + 1)k (2k + 2)!

(6.105)

k=0

n   (α + n + 1) (−1)k (λz)2κ+1 = .  (α + 2κ + 2) (2κ + 1)! (n − (2κ + 1)) k=0

Theorem 6.32. If z, λ ∈ C, n ∈ N, α > −1, |z| < 1, and i =

√ −1, then we have

 1  (α) Ln (iλz) + L(α) (−iλz) n 2 (α + 1)n = 1 Supercos 1 (−n; α + 1; λz) n! ∞ (α + 1)n  (−n)k (−1)k (λz)2k = n! (a + 1)k (2k + 1)!

(6.106)

k=0

=

n  k=0

 (α + n + 1) (−1)k (λz)2k .  (α + 2κ + 1) (2κ)! (n − 2κ)

Here we show the following result. Property 6.10. Let n ∈ N and α > −1. Then we have 1 Lsupersin1 (−n; α 1 Lsupercos 1 (−n; α

+ 1; 0) = 0,

+ 1; 0) =

1 Lsupertan1 (−n; α

(α + 1)n , n!

+ 1; 0) = 0,

(6.107) (6.108) (6.109)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and 1 Lsupersec 1 (−n; α

+ 1; 0) =

n! . (α + 1)n

(6.110)

We now present integral representations for the hypergeometric supersine and supercosine via Laguerre polynomials. Theorem 6.33 (Laplace transform). Let α, z, λ ∈ C, n ∈ N0 , and t ∈ [0, +∞). Then the Laplace transform of t λ−1 1 Lsupersin1 (−n; α + 1; zt) is as follows: L t

λ−1

 (α + 1)n  (λ) λ, −n z · λ · 2 Supersin1 , ; 1 Lsupersin1 (−n; α + 1; zt) = α+1 s n! s

(6.111)

where the Laplace transform of a function f is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(6.112)

0

Theorem 6.34. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and κ ∈ N, then 

  (c + 1)n α+β−1 ·z t α−1 (z − t)β−1 1 Lsupersin1 λt κ dt = n! 0

 ; −n, ακ , · · · , α+κ−1 κ κ × 1+κ Supersin1+κ , α+β α+β+κ−1 λz ; c + 1, κ , · · · , κ

1 B (α, β)

z

(6.113)

where λ is a constant. Theorem 6.35. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Lsupersin1 λt κ (z − t)s dt B (α, β) 0 (c + 1)n α+β−1 = ·z n!

 , βs , · · · , β+s−1 ; κ κ s s λzκ+s −n, ακ , · · · , α+κ−1 κ s × 1+κ+s Supersin1+κ+s κ+s , α+β+κ+s−1 ; (κ + s) c + 1, α+β κ+s , · · · , κ+s where λ is a constant.

(6.114)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

363

Theorem 6.36. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Lsupersin1 λ (z − t)s dt B (α, β) 0 (c + 1)n α+β−1 = ·z (6.115) n!

 β+s−1 −n, βs , β+1 ; s ,··· , s λzs , × 1+s Supersin1+s α+β α+β+1 α+β+s−1 ; c + 1, s , s , · · · , s where λ is a constant. Theorem 6.37. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then 

   t α−1 (1 − t)β−1 1 Lsupersin1 λ (z − t)s λt κ dt 0 

−n, ακ , · · · , α+κ−1 ; (c + 1)n κ = × 1+κ Supersin1+κ α+β+κ−1 λ , n! c + 1, α+β ; κ ,··· , κ

1 B (α, β)

1

(6.116)

where λ is a constant. Theorem 6.38. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Lsupersin1 λt κ (1 − t)s dt B (α, β) 0 (c + 1)n = (6.117) n!

 , βs , · · · , β+s−1 ; κ κ ss λ −n, ακ , · · · , α+κ−1 κ s × 1+κ+s Supersin1+κ+s κ+s , α+β+κ+s−1 c + 1, α+β ; (κ + s) κ+s , · · · , κ+s where λ is a constant. Theorem 6.39. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Lsupersin1 λ (1 − t)s dt B (α, β) 0 (c + 1)n = n! 

β+s−1 −n, βs , β+1 ; s ,··· , s λ , × 1+s Supersin1+s α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s where λ is a constant.

(6.118)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.40. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then 

  (c + 1)n α+β−1 ·2 (1 − t)α−1 (1 + t)β−1 1 Lsupersin1 λ (1 − t)κ dt = n! −1

 ; −n, ακ , · · · , α+κ−1 κ κ × 1+κ Supersin1+κ , α+β α+β+κ−1 λ2 c + 1, κ , · · · , ; κ

1 B (α, β)

1

(6.119)

where λ is a constant. Theorem 6.41. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  1   1 (1 − t)α−1 (1 + t)β−1 1 Lsupersin1 λ (1 − t)κ (1 + t)s dt B (α, β) −1 (c + 1)n α+β−1 = ·2 (6.120) n!

 , βs , · · · , β+s−1 ; κ κ s s λ2κ+s −n, ακ , · · · , α+κ−1 κ s × 1+κ+s Supersin1+κ+s κ+s , α+β+κ+s−1 c + 1, α+β ; (κ + s) κ+s , · · · , κ+s where λ is a constant. Theorem 6.42. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 (1 − t)α−1 (1 + t)β−1 1 Lsupersin1 λ (1 + t)s dt B (α, β) −1 (c + 1)n α+β−1 ·2 = n!

 β+s−1 −n, βs , β+1 ; s ,··· , s s × 1+s Supersin1+s λ2 , α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s

(6.121)

where λ is a constant. Theorem 6.43 (Laplace transform). Let α, z, λ ∈ C, n ∈ N0 , and t ∈ [0, +∞). Then the Laplace transform of t λ−1 1 Lsupercos 1 (−n; α + 1; zt) is as follows: 

(α + 1)  (λ) λ, −n z n ; · λ · 2 Supercos 1 , L t λ−1 1 Lsupercos 1 (−n; α + 1; zt) = α+1 s n! s

(6.122)

where the Laplace transform of a function f is  L [f (t)] = f (s) = 0

∞

e−st f (t) dt.

(6.123)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

365

Theorem 6.44. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and κ ∈ N, then 

  (c + 1)n α+β−1 t α−1 (z − t)β−1 1 Lsupercos 1 λt κ dt = ·z n! 0

 ; −n, ακ , · · · , α+κ−1 κ κ × 1+κ Supercos 1+κ , α+β α+β+κ−1 λz ; c + 1, κ , · · · , κ

1 B (α, β)

z

(6.124)

where λ is a constant. Theorem 6.45. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Lsupercos 1 λt κ (z − t)s dt B (α, β) 0 (c + 1)n α+β−1 = ·z n!

 β β+s−1 κ s s λzκ+s , , · · · , ; −n, ακ , · · · , α+κ−1 κ κ s s × 1+κ+s Supercos 1+κ+s κ+s , α+β+κ+s−1 c + 1, α+β , · · · , ; (κ + s) κ+s κ+s

(6.125)

where λ is a constant. Theorem 6.46. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Lsupercos 1 λ (z − t)s dt B (α, β) 0 (c + 1)n α+β−1 ·z = n!

 β+s−1 −n, βs , β+1 ; s ,··· , s s × 1+s Supercos 1+s λz , α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s

(6.126)

where λ is a constant. Theorem 6.47. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then 

   t α−1 (1 − t)β−1 1 Lsupercos 1 λ (z − t)s λt κ dt 0 

−n, ακ , · · · , α+κ−1 ; (c + 1)n κ = × 1+κ Supercos 1+κ α+β+κ−1 λ , n! ; c + 1, α+β κ ,··· , κ

1 B (α, β)

where λ is a constant.

1

(6.127)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.48. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Lsupercos 1 λt κ (1 − t)s dt B (α, β) 0 (c + 1)n = (6.128) n!

 , βs , · · · , β+s−1 ; κ κ ss λ −n, ακ , · · · , α+κ−1 κ s × 1+κ+s Supercos 1+κ+s κ+s , α+β+κ+s−1 ; (κ + s) c + 1, α+β κ+s , · · · , κ+s where λ is a constant. Theorem 6.49. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Lsupercos 1 λ (1 − t)s dt B (α, β) 0 (c + 1)n = n! 

β+s−1 −n, βs , β+1 ; s ,··· , s λ , × 1+s Supercos 1+s α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s

(6.129)

where λ is a constant. Theorem 6.50. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then 

  (c + 1)n α+β−1 ·2 (1 − t)α−1 (1 + t)β−1 1 Lsupercos 1 λ (1 − t)κ dt = n! −1

 −n, ακ , · · · , α+κ−1 ; κ κ × 1+κ Supercos 1+κ , α+β α+β+κ−1 λ2 ; c + 1, κ , · · · , κ

1 B (α, β)

1

(6.130)

where λ is a constant. Theorem 6.51. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  1   1 (1 − t)α−1 (1 + t)β−1 1 Lsupercos 1 λ (1 − t)κ (1 + t)s dt B (α, β) −1 (c + 1)n α+β−1 ·2 = (6.131) n!

 , βs , · · · , β+s−1 ; κ κ s s λ2κ+s −n, ακ , · · · , α+κ−1 κ s × 1+κ+s Supercos 1+κ+s κ+s , α+β α+β+κ+s−1 ; (κ + s) c + 1, κ+s , · · · , κ+s where λ is a constant.

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

367

Theorem 6.52. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 (1 − t)α−1 (1 + t)β−1 1 Lsupercos 1 λ (1 + t)s dt B (α, β) −1 (c + 1)n α+β−1 = ·2 n!

 β+s−1 −n, βs , β+1 ; s ,··· , s s × 1+s Supercos 1+s λ2 , α+β+1 c + 1, α+β , · · · , α+β+s−1 ; s , s s

(6.132)

where λ is a constant.

6.1.4 The hypergeometric superhyperbolic functions via Laguerre polynomials In this section, we propose the hypergeometric superhyperbolic functions via Laguerre polynomials. Definition 6.9. The hypergeometric superhyperbolic supersine via Laguerre polynomials is defined as + 1; z)  1  (α) Ln (z) − L(α) = (−z) n 2 (α + 1)n = 1 Supersinh1 (−n; α + 1; z) n! n  z2κ+1  (α + n + 1) , =  (α + 2κ + 1 + 1) (2κ + 1)! (n − (2κ + 1))

1 Lsupersinh1 (−n; α

(6.133)

k=0

where z ∈ C, n ∈ N, α > −1, and |z| < 1. Definition 6.10. The hypergeometric superhyperbolic supercosine via Laguerre polynomials is defined as + 1; z)  (α) L(α) n (z) + Ln (−z)

1 Lsupercosh1 (−n; α

=

1

2 (α + 1)n = 1 Supercosh1 (−n; α + 1; λz) n! n  z2k  (α + n + 1) , =  (α + 2κ + 1) (2κ)! (n − 2κ) k=0

where z ∈ C, n ∈ N, α > −1, and |z| < 1.

(6.134)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Definition 6.11. The hypergeometric superhyperbolic supertangent via Laguerre polynomials is defined as 1 Lsupertanh1 (−n; α

+ 1; z) =

+ 1; z) , Lsupercosh α + 1; z) (−n; 1 1 1 Lsupersinh1 (−n; α

(6.135)

where z ∈ C, n ∈ N, α > −1, and |z| < 1. Definition 6.12. The hypergeometric superhyperbolic supercotangent via Laguerre polynomials is defined as 1 Lsupercoth1 (−n; α

+ 1; z) =

+ 1; z) , 1 Lsupersinh1 (−n; α + 1; z)

1 Lsupercosh1 (−n; α

(6.136)

where z ∈ C, n ∈ N, α > −1, and |z| < 1. Definition 6.13. The hypergeometric superhyperbolic supersecant via Laguerre polynomials is defined as 1 Lsupersech1 (−n; α

+ 1; z) =

1 , Lsupercosh 1 1 (−n; α + 1; z)

(6.137)

where z ∈ C, n ∈ N, α > −1, and |z| < 1. Definition 6.14. The hypergeometric superhyperbolic supercosecant via Laguerre polynomials is defined as 1 Lsupercosech1 (−n; α

+ 1; z) =

1 , Lsupersinh 1 1 (−n; α + 1; z)

(6.138)

where z ∈ C, n ∈ N, α > −1, and |z| < 1. Definition 6.15. The hypergeometric superhyperbolic supersine 1 Lsupersinh1 (−n;α+1;z), hypergeometric superhyperbolic supercosine 1 Lsupercosh1 (−n; α + 1; z), hypergeometric superhyperbolic supertangent 1 Lsupertanh1 (−n; α + 1; z), hypergeometric superhyperbolic supercotangent 1 Lsupercoth1 (−n; α + 1; z), hypergeometric superhyperbolic supersecant 1 Lsupersech1 (−n; α + 1; z), and hypergeometric superhyperbolic supercosecant 1 Lsupercosec 1 (−n; α + 1; z) are called the hypergeometric superhyperbolic functions via Laguerre polynomials. At the moment, we get the following property. Property 6.11. Let z ∈ C, n ∈ N, α > −1, and |z| < 1. Then we have that 1 Lsupersinh1 (−n; α

+ 1; −z) = −1 Lsupersinh1 (−n; α + 1; z),

(6.139)

+ 1; −z) = 1 Lsupercosh1 (−n; α + 1; z) ,

(6.140)

+ 1; −z) = −1 Lsupertanh1 (−n; α + 1; z),

(6.141)

1 Lsupercosh1 (−n; α 1 Lsupertanh1 (−n; α

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

1 Lsupercoth1 (−n; α

369

+ 1; −z) = −1 Lsupercoth1 (−n; α + 1; z),

(6.142)

+ 1; −z) = 1 Lsupersech1 (−n; α + 1; z) ,

(6.143)

+ 1; −z) = −1 Lsupercosec 1 (−n; α + 1; z).

(6.144)

1 Lsupersech1 (−n; α

and 1 Lsupercosec 1 (−n; α

Moreover, we show that L(α) n (λz) =

n   (α + n + 1) (−λz)k  (α + k + 1) k! (n − k) k=0

n   (α + n + 1) (λz)2k =  (α + 2κ + 1) (2κ)! (n − 2κ)

+

k=0 n  k=0

(6.145)

 (α + n + 1) (λz)2κ+1  (α + 2κ + 2) (2κ + 1)! (n − (2κ + 1))

= 1 Lsupercosh1 (−n; α + 1; λz) + 1 Lsupersinh1 (−n; α + 1; λz) and L(α) n (−λz) = = −

n   (α + n + 1) k=0 n  k=0 n  k=0

(λz)k  (α + k + 1) k! (n − k)

 (α + n + 1) (λz)2k  (α + 2κ + 1) (2κ)! (n − 2κ)

(6.146)

 (α + n + 1) (λz)2κ+1  (α + 2κ + 2) (2κ + 1)! (n − (2κ + 1))

= 1 Lsupercosh1 (−n; α + 1; λz) − 1 Lsupersinh1 (−n; α + 1; λz) , where z ∈ C, k, n ∈ N, α > −1, and |z| < 1. At the moment, it is not difficult to show the following theorems. √ Theorem 6.53. If z ∈ C, k, n ∈ N, α > −1, |z| < 1, and i = −1, then we have L(α) n (λz) = 1 Lsupercosh1 (−n; α + 1; λz) + 1 Lsupersinh1 (−n; α + 1; λz)

(6.147)

L(α) n (−λz) = 1 Lsupercosh1 (−n; α + 1; λz) − 1 Lsupersinh1 (−n; α + 1; λz).

(6.148)

and

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

In this case, we have the following formulae. With the relations L(α) n (λz) =

n   (α + n + 1) (−λz)k  (α + k + 1) k! (n − k) k=0

n   (α + n + 1) (λz)2k =  (α + 2κ + 1) (2κ)! (n − 2κ)

(6.149)

k=0

n   (α + n + 1) (λz)2κ+1 +  (α + 2κ + 2) (2κ + 1)! (n − (2κ + 1)) k=0

and L(α) n (−λz) = =

n   (α + n + 1) k=0 n  k=0

−

n  k=0

(λz)k  (α + k + 1) k! (n − k)

 (α + n + 1) (λz)2k  (α + 2κ + 1) (2κ)! (n − 2κ)

(6.150)

 (α + n + 1) (λz)2κ+1 ,  (α + 2κ + 1 + 1) (2κ + 1)! (n − (2κ + 1))

we have that + 1; λz)   1 (α) Ln (λz) − L(α) = n (−λz) 2 n   (α + n + 1) (λz)2κ+1 = ,  (α + 2κ + 2) (2κ + 1)! (n − (2κ + 1))

(6.151)

+ 1; λz)   1 (α) Ln (λz) + L(α) = n (−λz) 2 n   (α + n + 1) (λz)2k = ,  (α + 2κ + 1) (2κ)! (n − 2κ)

(6.152)

1 Lsupersinh1 (−n; α

k=0

1 Lsupercosh1 (−n; α

k=0

1 Lsupertanh1 (−n; α

+ 1; λz) =

1 Lsupercoth1 (−n; α

+ 1; λz) =

+ 1; λz) , Lsupercosh α + 1; λz) (−n; 1 1

(6.153)

+ 1; λz) , 1 Lsupersinh1 (−n; α + 1; λz)

(6.154)

1 Lsupersinh1 (−n; α

1 Lsupercosh1 (−n; α

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

1 Lsupersech1 (−n; α

+ 1; λz) =

1 , 1 Lsupercosh1 (−n; α + 1; λz)

371

(6.155)

and 1 Lsupercosech1 (a; c; λz) =

where z ∈ C, n ∈ N, α > −1, |z| < 1, and i =

√

1 , 1 Lsupersinh1 (−n; α + 1; λz)

(6.156)

−1.

Property 6.12. Let n ∈ N and α > −1. Then we have 1 Lsupersinh1 (−n; α 1 Lsupercosh1 (−n; α

+ 1; 0) = 0,

+ 1; 0) =

1 Lsupertanh1 (−n; α

(α + 1)n , n!

+ 1; 0) = 0,

(6.157) (6.158) (6.159)

and 1 Lsupersech1 (−n; α

+ 1; 0) =

n! . (α + 1)n

(6.160)

Theorem 6.54. If z, λ ∈ C, n ∈ N, α > −1, and |z| < 1, then we have  1  (α) Ln (λz) − L(α) n (−λz) 2 (α + 1)n = 1 Supersinh1 (−n; α + 1; λz) n! ∞ (α + 1)n  (−n)k (λz)2k+1 = n! (a + 1)k (2k + 2)!

(6.161)

k=0

=

n  k=0

 (α + n + 1) (λz)2κ+1 .  (α + 2κ + 1 + 1) (2κ + 1)! (n − (2κ + 1))

Theorem 6.55. If z, λ ∈ C, n ∈ N, α > −1, and |z| < 1, then we have  1  (α) Ln (λz) + L(α) (−λz) n 2 (α + 1)n = 1 Supercosh1 (−n; α + 1; λz) n! ∞ (α + 1)n  (−n)k (λz)2k = n! (a + 1)k (2k + 1)! k=0

n   (α + n + 1) (λz)2k . =  (α + 2κ + 1) (2κ)! (n − 2κ) k=0

(6.162)

372

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Definition 6.16. Let ϕ ∈ C\Z and α > −1. The Laguerre type function is defined as L(α) ϕ (z) =

(α + 1)ϕ 1 F1 (−ϕ; α + 1; z). ϕ!

(6.163)

Based on this definition, we have the following results. Property 6.13. Let ϕ ∈ C\Z and α > −1. Then we have sin (ϕπ ) L(α) ϕ (z) = −



π

1

t −ϕ−1 (1 − t)α+ϕ ezt dt.

(6.164)

0

Proof. By means of the relation 1 1 sin (πϕ) · =−  (1 + ϕ)  (−ϕ) π

(6.165)

we have L(α) ϕ (z)

 1 (α + 1)ϕ  (α + 1) t −ϕ−1 (1 − t)α+ϕ ezt dt · ϕ!  (−ϕ)  (α + ϕ + 1) 0  1 1 1 · t −ϕ−1 (1 − t)α+ϕ ezt dt =  (1 + ϕ)  (−ϕ) 0  sin (πϕ) 1 −ϕ−1 t =− (1 − t)α+ϕ ezt dt. π 0

=

(6.166)

Property 6.14. Let ϕ ∈ C\Z and α > −1. Then we have L(α) ϕ (z) = −(α + 1)ϕ ·

sin (πϕ) π



∞

e−t t −ϕ−1 0 F1 (−; α + 1; zt)dt.

(6.167)

0

Proof. Using the relation 1 1 sin (πϕ) · =− ,  (1 + ϕ)  (−ϕ) π

(6.168)

L(α) ϕ (z)  ∞ (α + 1)ϕ 1 = · e−t t −ϕ−1 0 F1 (−; α + 1; zt)dt ϕ!  (−ϕ) 0  sin (πϕ) ∞ −t −ϕ−1 e t = −(α + 1)ϕ · 0 F1 (−; α + 1; zt)dt. π 0

(6.169)

we obtain

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

373

When ϕ = n ∈ N, we have (α) L(α) ϕ (z) = Ln (z) ,

(6.170)

which are the Laguerre polynomials. Theorem 6.56. If ϕ ∈ C\Z and α > −1, then we have 1 Lsupersin1 (−ϕ; α

+ 1; z) = −

sin (ϕπ) π



1

t −ϕ−1 (1 − t)α+ϕ sin (zt) dt.

(6.171)

0

Proof. We have that 1 Lsupersin1 (−ϕ; α

+ 1; z)

 (α) L(α) − L (iz) (−iz) ϕ ϕ   1   sin (ϕπ) −ϕ−1 α+ϕ izt −izt − e −e dt t (1 − t) π 0     1 izt sin (ϕπ) 1 −ϕ−1 e − e−izt dt t =− (1 − t)α+ϕ π 2i 0  1 sin (ϕπ) t −ϕ−1 (1 − t)α+ϕ sin (zt) dt, =− π 0 1 = 2i 1 = 2i



(6.172)

which is derived from the representations sin (ϕπ ) L(α) ϕ (iz) = − π

sin (ϕπ) L(α) ϕ (−iz) = − π



1

t −ϕ−1 (1 − t)α+ϕ eizt dt,

(6.173)

t −ϕ−1 (1 − t)α+ϕ e−izt dt,

(6.174)

 1  (α) Lϕ (iz) − L(α) ϕ (−iz) . 2i

(6.175)

0



1

0

and 1 Lsupersin1 (−ϕ; α

+ 1; z) =

Theorem 6.57. If ϕ ∈ C\Z and α > −1, then we have 1 Lsupercos 1 (−ϕ; α + 1; z) = −

sin (ϕπ) π

 0

1

t −ϕ−1 (1 − t)α+ϕ cos (zt) dt.

(6.176)

374

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Proof. We have the integral representation + 1; z)  (α) L(α) = ϕ (iz) + Lϕ (−iz) 2     sin (ϕπ ) 1 −ϕ−1 1 − = t (1 − t)α+ϕ eizt + e−izt dt 2 π 0     sin (ϕπ ) 1 −ϕ−1 α+ϕ 1 izt −izt e +e dt t =− (1 − t) π 2 0  sin (ϕπ ) 1 −ϕ−1 t =− (1 − t)α+ϕ cos (zt) dt, π 0

1 Lsupercosin1 (−ϕ; α

1

where sin (ϕπ ) L(α) ϕ (iz) = − π

sin (ϕπ) L(α) ϕ (−iz) = −



1

(6.177)

t −ϕ−1 (1 − t)α+ϕ eizt dt,

(6.178)

t −ϕ−1 (1 − t)α+ϕ e−izt dt,

(6.179)

 1  (α) . Ln (iz) + L(α) (−iz) n 2

(6.180)

0



π

1

0

and 1 Lsupercos 1 (−n; α + 1; z) =

Theorem 6.58. If ϕ ∈ C\Z and α > −1, then we have  sin (ϕπ) 1 −ϕ−1 Lsupersinh t α + 1; z) = − (−ϕ; (1 − t)α+ϕ sinh (zt) dt. 1 1 π 0

(6.181)

Proof. We have that + 1; z)  (α) L(α) = ϕ (z) − Lϕ (−z) 2     1 sin (ϕπ) 1 −ϕ−1 = t − (1 − t)α+ϕ ezt − e−zt dt 2 π 0   1  sin (ϕπ) 1  zt =− e − e−zt dt t −ϕ−1 (1 − t)α+ϕ π 2 0  1 sin (ϕπ) =− t −ϕ−1 (1 − t)α+ϕ sinh (zt) dt π 0

1 Lsupersinh1 (−ϕ; α

1

since L(α) ϕ (z) = −

sin (ϕπ) π

 0

1

t −ϕ−1 (1 − t)α+ϕ ezt dt,

(6.182)

(6.183)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

L(α) ϕ (−z) = −

sin (ϕπ) π



1

t −ϕ−1 (1 − t)α+ϕ e−zt dt,

375

(6.184)

0

and 1 Lsupersinh1 (−ϕ; α

+ 1; z) = −

 1  (α) Lϕ (z) − L(α) ϕ (−z) . 2

(6.185)

Theorem 6.59. If ϕ ∈ C\Z and α > −1, then we have sin (ϕπ) 1 Lsupercosh1 (−ϕ; α + 1; z) = − π



1

t −ϕ−1 (1 − t)α+ϕ cosh (zt) dt.

(6.186)

0

Proof. We have + 1; z)  (α) L(α) = ϕ (z) + Lϕ (−z) 2     sin (ϕπ) 1 −ϕ−1 1 − = t (1 − t)α+ϕ ezt + e−zt dt 2 π 0   1   sin (ϕπ) −ϕ−1 α+ϕ 1 zt −zt e +e dt t =− (1 − t) π 2 0  1 sin (ϕπ) =− t −ϕ−1 (1 − t)α+ϕ cosh (zt) dt, π 0

1 Lsupercosinh1 (−ϕ; α

1

(6.187)

from which it follows that sin (ϕπ) L(α) ϕ (z) = − π

L(α) ϕ (−z) = −

sin (ϕπ) π



1

t −ϕ−1 (1 − t)α+ϕ ezt dt,

(6.188)

t −ϕ−1 (1 − t)α+ϕ e−zt dt,

(6.189)

0



1

0

and 1 Lsupercosh1 (−n; α

+ 1; z) =

 1  (α) Ln (z) + L(α) n (−z) . 2

(6.190)

Theorem 6.60. If ϕ ∈ C\Z and α > −1, then we have 1 Lsupersin1 (−ϕ; α

+ 1; z) =

(α + 1)ϕ 1 Supersin1 (−ϕ; α + 1; z). ϕ!

(6.191)

376

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Proof. We have + 1; z)   1 (α) L(α) = ϕ (iz) − Lϕ (−iz) 2i 1 (α + 1)ϕ = (1 F1 (−ϕ; α + 1; iz) − 1 F1 (−ϕ; α + 1; −iz)) 2i ϕ!  (α + 1)ϕ 1 = (1 F1 (−ϕ; α + 1; iz) − 1 F1 (−ϕ; α + 1; −iz)) ϕ! 2i (α + 1)ϕ = 1 Supersin1 (−ϕ; α + 1; z) , ϕ!

1 Lsupersin1 (−ϕ; α

(6.192)

which is derived from L(α) ϕ (iz) = L(α) ϕ (−iz) = 1 Supersin1 (−ϕ; α

+ 1; z) =

(α + 1)ϕ 1 F1 (−ϕ; α + 1; iz), ϕ!

(6.193)

(α + 1)ϕ 1 F1 (−ϕ; α + 1; −iz), ϕ!

(6.194)

1 (1 F1 (−ϕ; α + 1; iz) + 1 F1 (−ϕ; α + 1; −iz)) , 2i

and 1 Lsupersin1 (−ϕ; α

+ 1; z) = −

 1  (α) Lϕ (iz) − L(α) ϕ (−iz) . 2i

(6.195)

(6.196)

Theorem 6.61. If ϕ ∈ C\Z and α > −1, then we have 1 Lsupercos 1 (−ϕ; α

+ 1; z) =

(α + 1)ϕ 1 Supercos 1 (−ϕ; α + 1; z). ϕ!

(6.197)

Proof. We have + 1; z)  1  (α) Lϕ (iz) + L(α) (−iz) ϕ 2 + 1) 1 (α ϕ (1 F1 (−ϕ; α + 1; iz) + 1 F1 (−ϕ; α + 1; −iz)) 2 ϕ!  (α + 1)ϕ 1 (1 F1 (−ϕ; α + 1; iz) + 1 F1 (−ϕ; α + 1; −iz)) ϕ! 2 (α + 1)ϕ 1 Supercos 1 (−ϕ; α + 1; z) ϕ!

1 Lsupercos 1 (−ϕ; α

= = = =

(6.198)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

377

since L(α) ϕ (iz) = L(α) ϕ (−iz) = 1 Supercos 1 (−ϕ; α

+ 1; z) =

(α + 1)ϕ 1 F1 (−ϕ; α + 1; iz), ϕ!

(6.199)

(α + 1)ϕ 1 F1 (−ϕ; α + 1; −iz), ϕ!

(6.200)

1 (1 F1 (−ϕ; α + 1; iz) + 1 F1 (−ϕ; α + 1; −iz)) , 2

and 1 Lsupercos 1 (−ϕ; α

+ 1; z) =

 1  (α) Lϕ (iz) + L(α) . (−iz) ϕ 2

(6.201)

(6.202)

Theorem 6.62. If z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1, then we have 1 Lsupertan1 (−ϕ; α

+ 1; z) = 1 Lsupertan1 (−ϕ; α + 1; z) .

(6.203)

Proof. We have 1 Lsupertan1 (−ϕ; α

+ 1; z)

1 Lsupersin1 (−ϕ; α + 1; z) = 1 Lsupercos 1 (−ϕ; α + 1; z)

=

(α+1)ϕ ϕ! 1 Supersin1 (−ϕ; α + 1; z) (α+1)ϕ ϕ! 1 Supercos 1 (−ϕ; α + 1; z)

=

+ 1; z) 1 Supercos 1 (−ϕ; α + 1; z)

(6.204)

1 Supersin1 (−ϕ; α

= 1 Supertan1 (−ϕ; α + 1; z) , where z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1. Theorem 6.63. If z ∈ C, ϕ ∈ C\Z, α > −1 and |z| < 1, then we have 1 Lsupercot 1 (−ϕ; α

+ 1; z) = 1 Supercot 1 (−ϕ; α + 1; z) .

(6.205)

Proof. We have + 1; z) 1 Lsupercos 1 (−ϕ; α + 1; z) = 1 Lsupersin1 (−ϕ; α + 1; z)

1 Lsupercot 1 (−ϕ; α

=

(α+1)ϕ ϕ! 1 Supercos 1 (−ϕ; α + 1; z) (α+1)ϕ ϕ! 1 Supersin1 (−ϕ; α + 1; z)

(6.206)

378

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

=

+ 1; z) 1 Supersin1 (−ϕ; α + 1; z)

1 Supercos 1 (−ϕ; α

= 1 Supercot 1 (−ϕ; α + 1; z) , where z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1. Theorem 6.64. If z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1, then we have 1 Lsupersec 1 (−ϕ; α

+ 1; z) =

ϕ! 1 Supersec 1 (−ϕ; α + 1; z). (α + 1)ϕ

(6.207)

Proof. It is clear that 1 Lsupersec 1 (−ϕ; α

= =

+ 1; z)

1 1 Lsupercos 1 (−ϕ; α

+ 1; z)

1 (α+1)ϕ ϕ! 1 Supercos 1 (−ϕ; α

+ 1; z)

(6.208)

ϕ! 1 (α + 1)ϕ 1 Supercos 1 (−ϕ; α + 1; z) ϕ! = 1 Supersec 1 (−n; α + 1; z), (α + 1)ϕ

=

where z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1. Theorem 6.65. If z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1, then we have 1 Lsupercosec 1 (−ϕ; α

+ 1; z) =

ϕ! 1 Supercosec 1 (−ϕ; α + 1; z). (α + 1)ϕ

(6.209)

Proof. We have 1 Lsupercosec 1 (−ϕ; α

= =

+ 1; z)

1 Lsupersin α + 1; z) (−ϕ; 1 1 1 (α+1)ϕ ϕ! 1 Supersin1 (−ϕ; α

+ 1; z)

ϕ! 1 (α + 1)ϕ 1 Supersin1 (−ϕ; α + 1; z) ϕ! = 1 Supercosec 1 (−n; α + 1; z), (α + 1)ϕ

=

where z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1.

(6.210)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

379

Theorem 6.66. If ϕ ∈ C\Z and α > −1, then we have (α + 1)ϕ 1 Supersinh1 (−ϕ; α + 1; z). ϕ!

(6.211)

+ 1; z)   1 (α) Lϕ (z) − L(α) ϕ (−z) 2 1 (α + 1)ϕ (1 F1 (−ϕ; α + 1; z) − 1 F1 (−ϕ; α + 1; −z)) 2 ϕ!  (α + 1)ϕ 1 (1 F1 (−ϕ; α + 1; z) − 1 F1 (−ϕ; α + 1; −z)) ϕ! 2 (α + 1)ϕ 1 Supersinh1 (−ϕ; α + 1; z) ϕ!

(6.212)

1 Lsupersinh1 (−ϕ; α

+ 1; z) =

Proof. We have 1 Lsupersinh1 (−ϕ; α

= = = = since

L(α) ϕ (z) = L(α) ϕ (−z) = 1 Supersinh1 (−ϕ; α

+ 1; z) =

(α + 1)ϕ 1 F1 (−ϕ; α + 1; z), ϕ!

(6.213)

(α + 1)ϕ 1 F1 (−ϕ; α + 1; −z), ϕ!

(6.214)

1 (1 F1 (−ϕ; α + 1; z) − 1 F1 (−ϕ; α + 1; −z)) , 2

and 1 Lsupersinh1 (−ϕ; α

+ 1; z) =

 1  (α) . Lϕ (z) − L(α) (−z) ϕ 2

(6.215)

(6.216)

Theorem 6.67. If ϕ ∈ C\Z and α > −1, then we have 1 Lsupercosh1 (−ϕ; α

+ 1; z) = −

(α + 1)ϕ 1 Supercosh1 (−ϕ; α + 1; z). ϕ!

(6.217)

Proof. We have + 1; z)   1 (α) Lϕ (z) + L(α) = ϕ (−z) 2 1 (α + 1)ϕ = (1 F1 (−ϕ; α + 1; z) + 1 F1 (−ϕ; α + 1; −z)) 2 ϕ!

1 Lsupercosh1 (−ϕ; α

(6.218)

380

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions  (α + 1)ϕ 1 (1 F1 (−ϕ; α + 1; z) + 1 F1 (−ϕ; α + 1; −z)) ϕ! 2 (α + 1)ϕ = 1 Supercosh1 (−ϕ; α + 1; z) ϕ!

=

since L(α) ϕ (z) = L(α) ϕ (−z) = 1 Supercosh1 (−ϕ; α

+ 1; z) =

(α + 1)ϕ 1 F1 (−ϕ; α + 1; z), ϕ!

(6.219)

(α + 1)ϕ 1 F1 (−ϕ; α + 1; −z), ϕ!

(6.220)

1 (1 F1 (−ϕ; α + 1; z) + 1 F1 (−ϕ; α + 1; −z)) , 2

(6.221)

and 1 Lsupercosh1 (−ϕ; α

+ 1; z) =

 1  (α) . Lϕ (z) + L(α) (−z) ϕ 2

(6.222)

Theorem 6.68. If z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1, then we have 1 Lsupertanh1 (−ϕ; α

+ 1; z) = 1 Lsupertanh1 (−ϕ; α + 1; z) .

(6.223)

Proof. We have 1 Lsupertanh1 (−ϕ; α

+ 1; z)

=

+ 1; z) 1 Lsupercosh1 (−ϕ; α + 1; z)

=

(α+1)ϕ ϕ! 1 Supersinh1 (−ϕ; α + 1; z) (α+1)ϕ ϕ! 1 Supercosh1 (−ϕ; α + 1; z)

=

1 Lsupersinh1 (−ϕ; α

(6.224)

+ 1; z) Supercos α + 1; z) 1 1 (−ϕ;

1 Supersinh1 (−ϕ; α

= 1 Supertanh1 (−ϕ; α + 1; z) , where z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1. Theorem 6.69. If z ∈ C, ϕ ∈ C\Z, α > −1 and |z| < 1, then we have 1 Lsupercoth1 (−ϕ; α

+ 1; z) = 1 Supercoth1 (−ϕ; α + 1; z) .

(6.225)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

381

Proof. We have 1 Lsupercoth1 (−ϕ; α

= = =

+ 1; z)

1 Lsupercosh1 (−ϕ; α

+ 1; z)

1 Lsupersinh1 (−ϕ; α + 1; z) (α+1)ϕ ϕ! 1 Supercosh1 (−ϕ; α + 1; z) (α+1)ϕ ϕ! 1 Supersinh1 (−ϕ; α

+ 1; z)

(6.226)

+ 1; z) 1 Supersinh1 (−ϕ; α + 1; z)

1 Supercosh1 (−ϕ; α

= 1 Supercoth1 (−ϕ; α + 1; z) , where z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1. Theorem 6.70. If z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1, then we have 1 Lsupersech1 (−ϕ; α

+ 1; z) =

ϕ! 1 Supersech1 (−ϕ; α + 1; z). (α + 1)ϕ

(6.227)

Proof. We obtain + 1; z) 1 = 1 Lsupercosh1 (−ϕ; α + 1; z) 1 = (α+1) ϕ ϕ! 1 Supercosh1 (−ϕ; α + 1; z)

1 Lsupersech1 (−ϕ; α

(6.228)

ϕ! 1 (α + 1)ϕ 1 Supercosh1 (−ϕ; α + 1; z) ϕ! = 1 Supersech1 (−n; α + 1; z), (α + 1)ϕ

=

where z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1. Theorem 6.71. If z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1, then we have 1 Lsupercosech1 (−ϕ; α

+ 1; z) =

ϕ! 1 Supercosech1 (−ϕ; α + 1; z). (α + 1)ϕ

Proof. We have + 1; z) 1 = 1 Lsupersinh1 (−ϕ; α + 1; z)

1 Lsupercosech1 (−ϕ; α

(6.229)

382

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

=

1 (α+1)ϕ ϕ! 1 Supersinh1 (−ϕ; α

+ 1; z)

(6.230)

ϕ! 1 (α + 1)ϕ 1 Supersinh1 (−ϕ; α + 1; z) ϕ! = 1 Supercosech1 (−n; α + 1; z), (α + 1)ϕ

=

where z ∈ C, ϕ ∈ C\Z, α > −1, and |z| < 1. Theorem 6.72. If ϕ ∈ C\Z and α > −1, then we have + 1; z)  sin + 1) (ϕπ) ∞ −t −ϕ−1 (α ϕ e t =− 1 Supersin1 (−ϕ; α + 1; z)dt. π 0

1 Lsupersin1 (−ϕ; α

(6.231)

Proof. We have + 1; z)   1 (α) L(α) = − L (iz) (−iz) ϕ ϕ 2i (α + 1)ϕ sin (ϕπ ) =− π   ∞ 1 −t −ϕ−1 e t × (0 F1 (−; α + 1; izt) − 0 F1 (−; α + 1; −izt)) dt 2i 0  (α + 1)ϕ sin (ϕπ ) ∞ −t −ϕ−1 =− e t 0 Supersin1 (−ϕ; α + 1; z)dt, π 0

1 Lsupersin1 (−ϕ; α

(6.232)

which follows from the equations  (α + 1)ϕ sin (ϕπ) ∞ −t −ϕ−1 e t 0 F1 (−; α + 1; izt)dt, π 0  (α + 1)ϕ sin (ϕπ) ∞ −t −ϕ−1 (α) Lϕ (−iz) = − e t 0 F1 (−; α + 1; −izt)dt, π 0 L(α) ϕ (iz) = −

0 Supersin1 (−ϕ; α

+ 1; z) =

1 (0 F1 (−; α + 1; izt) − 0 F1 (−; α + 1; −izt)) , 2i

(6.233) (6.234) (6.235)

and 1 Lsupersin1 (−ϕ; α + 1; z) =

 1  (α) . Lϕ (iz) − L(α) (−iz) ϕ 2i

(6.236)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

383

Theorem 6.73. If ϕ ∈ C\Z and α > −1, then we have + 1; z)  (α + 1)ϕ sin (ϕπ) ∞ −t −ϕ−1 e t =− 1 Supercos 1 (−ϕ; α + 1; z)dt. π 0

1 Lsupercos 1 (−ϕ; α

(6.237)

Proof. We have + 1; z)   1 (α) Lϕ (iz) + L(α) = ϕ (−iz) 2   (α + 1)ϕ sin (ϕπ ) ∞ −t −ϕ−1 1 =− e t (0 F1 (−; α + 1; izt) + 0 F1 (−; α + 1; −izt)) dt π 2 0  ∞ (α + 1)ϕ sin (ϕπ ) =− e−t t −ϕ−1 0 Supercos 1 (−ϕ; α + 1; z)dt π 0 (6.238)

1 Lsupercos 1 (−ϕ; α

since  (α + 1)ϕ sin (ϕπ) ∞ −t −ϕ−1 e t 0 F1 (−; α + 1; izt)dt, π 0  (α + 1)ϕ sin (ϕπ) ∞ −t −ϕ−1 (α) Lϕ (−iz) = − e t 0 F1 (−; α + 1; −izt)dt, π 0 L(α) ϕ (iz) = −

0 Supercos 1 (−ϕ; α

+ 1; z) =

1 (0 F1 (−; α + 1; izt) + 0 F1 (−; α + 1; −izt)) , 2

and 1 Lsupercos 1 (−ϕ; α

+ 1; z) =

 1  (α) Lϕ (iz) + L(α) ϕ (−iz) . 2

(6.239) (6.240) (6.241)

(6.242)

Theorem 6.74. If ϕ ∈ C\Z and α > −1, then we have + 1; z)  (α + 1)ϕ sin (ϕπ) ∞ −t −ϕ−1 e t =− 1 Supersinh1 (−ϕ; α + 1; z)dt. π 0

1 Lsupersinh1 (−ϕ; α

Proof. We have + 1; z)   1 (α) Lϕ (z) − L(α) = ϕ (−z) 2

1 Lsupersinh1 (−ϕ; α

(6.243)

384

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions   (α + 1)ϕ sin (ϕπ ) ∞ −t −ϕ−1 1 e t (0 F1 (−; α + 1; izt) − 0 F1 (−; α + 1; −izt)) dt π 2 0  (α + 1)ϕ sin (ϕπ ) ∞ −t −ϕ−1 =− e t 0 Supersinh1 (−ϕ; α + 1; z)dt, π 0

=−

(6.244) which is derived from  (α + 1)ϕ sin (ϕπ) ∞ −t −ϕ−1 e t 0 F1 (−; α + 1; zt)dt, π 0  (α + 1)ϕ sin (ϕπ) ∞ −t −ϕ−1 (α) Lϕ (−z) = − e t 0 F1 (−; α + 1; −zt)dt, π 0 L(α) ϕ (z) = −

0 Supersinh1 (−ϕ; α

+ 1; z) =

1 (0 F1 (−; α + 1; zt) − 0 F1 (−; α + 1; −zt)) , 2

and 1 Lsupersinh1 (−ϕ; α + 1; z) =

 1  (α) . Lϕ (z) − L(α) (−z) ϕ 2

(6.245) (6.246) (6.247)

(6.248)

Theorem 6.75. If ϕ ∈ C\Z and α > −1, then we have + 1; z)  (α + 1)ϕ sin (ϕπ) ∞ −t −ϕ−1 e t =− 1 Supercosh1 (−ϕ; α + 1; z)dt. π 0

1 Lsupercosh1 (−ϕ; α

(6.249)

Proof. We have + 1; z)  1  (α) Lϕ (z) + L(α) = (−z) ϕ 2   (α + 1)ϕ sin (ϕπ ) ∞ −t −ϕ−1 1 =− e t (0 F1 (−; α + 1; izt) + 0 F1 (−; α + 1; −izt)) dt π 2 0  ∞ (α + 1)ϕ sin (ϕπ ) =− e−t t −ϕ−1 0 Supercos 1 (−ϕ; α + 1; z)dt, π 0 (6.250)

1 Lsupercosh1 (−ϕ; α

where  + 1)ϕ sin (ϕπ) ∞ −t −ϕ−1 e t 0 F1 (−; α + 1; zt)dt, π 0  (α + 1)ϕ sin (ϕπ) ∞ −t −ϕ−1 L(α) e t = − (−z) 0 F1 (−; α + 1; −zt)dt, ϕ π 0 (α L(α) ϕ (z) = −

(6.251) (6.252)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

0 Supercosh1 (−ϕ; α

+ 1; z) =

1 (0 F1 (−; α + 1; zt) + 0 F1 (−; α + 1; −zt)) , 2

and 1 Lsupercosh1 (−ϕ; α

+ 1; z) =

 1  (α) Lϕ (z) + L(α) ϕ (−z) . 2

385

(6.253)

(6.254)

We now consider the following properties. Property 6.15. Let n ∈ N and α > −1. Then we have + 1; z) = −i 1 Lsupersin1 (−n; α + 1; iz),

(6.255)

+ 1; z) = 1 Lsupercos 1 (−n; α + 1; iz) ,

(6.256)

+ 1; z) = −i 1 Lsupertanh1 (−n; α + 1; iz),

(6.257)

1 Lsupercoth1 (−n; α

+ 1; z) = i 1 Lsupercoth1 (−n; α + 1; iz),

(6.258)

1 Lsupersech1 (−n; α

+ 1; z) = 1 Lsupersech1 (−n; α + 1; iz),

(6.259)

+ 1; z) = i1 Lsupercosech1 (−n; α + 1; iz).

(6.260)

1 Lsupersinh1 (−n; α

1 Lsupercosh1 (−n; α 1 Lsupertanh1 (−n; α

and 1 Lsupercosech1 (−n; α

Property 6.16. Let ϕ ∈ C\Z and α > −1. Then we have + 1; z) = −i 1 Lsupersin1 (−ϕ; α + 1; iz),

(6.261)

+ 1; z) = 1 Lsupercos 1 (−ϕ; α + 1; iz) ,

(6.262)

+ 1; z) = −i 1 Lsupertanh1 (−ϕ; α + 1; iz),

(6.263)

1 Lsupercoth1 (−ϕ; α

+ 1; z) = i 1 Lsupercoth1 (−ϕ; α + 1; iz),

(6.264)

1 Lsupersech1 (−ϕ; α

+ 1; z) = 1 Lsupersech1 (−ϕ; α + 1; iz),

(6.265)

+ 1; z) = i1 Lsupercosech1 (−ϕ; α + 1; iz).

(6.266)

1 Lsupersinh1 (−ϕ; α

1 Lsupercosh1 (−ϕ; α 1 Lsupertanh1 (−ϕ; α

and 1 Lsupercosech1 (−ϕ; α

We consider that integral representations for the hypergeometric superhyperbolic supersine and supercosine via Laguerre polynomials. Theorem 6.76 (Laplace transform). Let α, z, λ ∈ C, n ∈ N0 , and t ∈ [0, +∞). Then the Laplace transform of t λ−1 1 Lsupersinh1 (−n; α + 1; zt) is as follows: 

(α + 1)  (λ) λ, −n z n λ−1 ; · λ · 2 Supersinh1 , L t 1 Lsupersinh1 (−n; α + 1; zt) = α+1 s n! s

(6.267)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

where the Laplace transform of a function f is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(6.268)

0

It is not difficult to prove the following theorems. Theorem 6.77. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and κ ∈ N, then  z   1 (c + 1)n α+β−1 ·z t α−1 (z − t)β−1 1 Lsupersinh1 λt κ dt = B (α, β) 0 n!

 (6.269) ; −n, ακ , · · · , α+κ−1 κ κ λz × 1+κ Supersinh1+κ , α+β+κ−1 ; c + 1, α+β κ ,··· , κ where λ is a constant. Theorem 6.78. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Lsupersinh1 λt κ (z − t)s dt B (α, β) 0 (c + 1)n α+β−1 ·z = (6.270) n!

 β β+s−1 , s , · · · , s ; κ κ s s λzκ+s −n, ακ , · · · , α+κ−1 κ × 1+κ+s Supersinh1+κ+s κ+s , α+β+κ+s−1 c + 1, α+β ; (κ + s) κ+s , · · · , κ+s where λ is a constant. Theorem 6.79. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Lsupersinh1 λ (z − t)s dt B (α, β) 0 (c + 1)n α+β−1 ·z = (6.271) n!

 β+s−1 −n, βs , β+1 ; s ,··· , s λzs , × 1+s Supersinh1+s α+β α+β+1 α+β+s−1 ; c + 1, s , s , · · · , s where λ is a constant. Theorem 6.80. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then  1    1 t α−1 (1 − t)β−1 1 Lsupersinh1 λ (z − t)s λt κ dt B (α, β) 0 

(6.272) ; −n, ακ , · · · , α+κ−1 (c + 1)n κ = × 1+κ Supersinh1+κ α+β+κ−1 λ , n! ; c + 1, α+β κ ,··· , κ where λ is a constant.

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

387

Theorem 6.81. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Lsupersinh1 λt κ (1 − t)s dt B (α, β) 0 (c + 1)n = (6.273) n!

 , βs , · · · , β+s−1 ; κ κ ss λ −n, ακ , · · · , α+κ−1 κ s × 1+κ+s Supersinh1+κ+s κ+s , α+β+κ+s−1 ; (κ + s) c + 1, α+β κ+s , · · · , κ+s where λ is a constant. Theorem 6.82. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Lsupersinh1 λ (1 − t)s dt B (α, β) 0 (c + 1)n = n! 

β+s−1 −n, βs , β+1 ; s ,··· , s λ , × 1+s Supersinh1+s α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s

(6.274)

where λ is a constant. Theorem 6.83. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then 

  (c + 1)n α+β−1 ·2 (1 − t)α−1 (1 + t)β−1 1 Lsupersinh1 λ (1 − t)κ dt = n! −1

 −n, ακ , · · · , α+κ−1 ; κ κ × 1+κ Supersinh1+κ , α+β α+β+κ−1 λ2 ; c + 1, κ , · · · , κ

1 B (α, β)

1

(6.275)

where λ is a constant. Theorem 6.84. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  1   1 (1 − t)α−1 (1 + t)β−1 1 Lsupersinh1 λ (1 − t)κ (1 + t)s dt B (α, β) −1 (c + 1)n α+β−1 ·2 = (6.276) n!

 , βs , · · · , β+s−1 ; κ κ s s λ2κ+s −n, ακ , · · · , α+κ−1 κ s × 1+κ+s Supersinh1+κ+s κ+s , α+β α+β+κ+s−1 ; (κ + s) c + 1, κ+s , · · · , κ+s where λ is a constant.

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Theorem 6.85. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 (1 − t)α−1 (1 + t)β−1 1 Lsupersinh1 λ (1 + t)s dt B (α, β) −1 (c + 1)n α+β−1 = ·2 n!

 β+s−1 −n, βs , β+1 ; s ,··· , s s × 1+s Supersinh1+s λ2 , α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s

(6.277)

where λ is a constant. Theorem 6.86 (Laplace transform). Let α, z, λ ∈ C, n ∈ N0 , and t ∈ [0, +∞). Then the Laplace transform of t λ−1 1 Lsupercosh1 (−n; α + 1; zt) is as follows: 

(α + 1)  (λ) λ, −n z n · λ · 2 Supercosh1 , ; L t λ−1 1 Lsupercosh1 (−n; α + 1; zt) = α+1 s n! s

(6.278)

where the Laplace transform of a function f is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(6.279)

0

In this case, we can derive the following theorems: Theorem 6.87. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and κ ∈ N, then  z   1 (c + 1)n α+β−1 ·z t α−1 (z − t)β−1 1 Lsupercosh1 λt κ dt = B (α, β) 0 n!

 (6.280) ; −n, ακ , · · · , α+κ−1 κ κ λz × 1+κ Supercosh1+κ , α+β+κ−1 ; c + 1, α+β κ ,··· , κ where λ is a constant. Theorem 6.88. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Lsupercosh1 λt κ (z − t)s dt B (α, β) 0 (c + 1)n α+β−1 ·z = (6.281) n!

 β+s−1 α α+κ−1 β −n, κ , · · · , κ , s , · · · , s ; κ κ s s λzκ+s × 1+κ+s Supercosh1+κ+s κ+s , α+β+κ+s−1 ; (κ + s) c + 1, α+β κ+s , · · · , κ+s where λ is a constant.

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

389

Theorem 6.89. If Re (α) > 0, Re (β) > 0, Re (c) > 0, |z| < 1, Re (c + 1) > −n, and s ∈ N, then we have  z   1 t α−1 (z − t)β−1 1 Lsupercosh1 λ (z − t)s dt B (α, β) 0 (c + 1)n α+β−1 = ·z (6.282) n!

 β+s−1 −n, βs , β+1 ; s ,··· , s λzs , × 1+s Supercosh1+s α+β α+β+1 α+β+s−1 ; c + 1, s , s , · · · , s where λ is a constant. Theorem 6.90. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then 

   t α−1 (1 − t)β−1 1 Lsupercosh1 λ (z − t)s λt κ dt 0 

−n, ακ , · · · , α+κ−1 ; (c + 1)n κ = × 1+κ Supercosh1+κ α+β+κ−1 λ , n! c + 1, α+β ; κ ,··· , κ

1 B (α, β)

1

(6.283)

where λ is a constant. Theorem 6.91. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Lsupercosh1 λt κ (1 − t)s dt B (α, β) 0 (c + 1)n = (6.284) n!

 , βs , · · · , β+s−1 ; κ κ ss λ −n, ακ , · · · , α+κ−1 κ s × 1+κ+s Supercosh1+κ+s κ+s , α+β+κ+s−1 c + 1, α+β ; (κ + s) κ+s , · · · , κ+s where λ is a constant. Theorem 6.92. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 t α−1 (1 − t)β−1 1 Lsupercosh1 λ (1 − t)s dt B (α, β) 0 (c + 1)n = n! 

β+s−1 −n, βs , β+1 ; s ,··· , s λ , × 1+s Supercosh1+s α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s where λ is a constant.

(6.285)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.93. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then  1   1 (c + 1)n α+β−1 ·2 (1 − t)α−1 (1 + t)β−1 1 Lsupercosh1 λ (1 − t)κ dt = B (α, β) −1 n!

 (6.286) ; −n, ακ , · · · , α+κ−1 κ κ × 1+κ Supercosh1+κ , α+β+κ−1 λ2 ; c + 1, α+β κ ,··· , κ where λ is a constant. Theorem 6.94. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  1   1 (1 − t)α−1 (1 + t)β−1 1 Lsupercosh1 λ (1 − t)κ (1 + t)s dt B (α, β) −1 (c + 1)n α+β−1 = ·2 (6.287) n!

 , βs , · · · , β+s−1 ; κ κ s s λ2κ+s −n, ακ , · · · , α+κ−1 κ s × 1+κ+s Supercosh1+κ+s κ+s , α+β+κ+s−1 c + 1, α+β ; (κ + s) κ+s , · · · , κ+s where λ is a constant. Theorem 6.95. If Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 (1 − t)α−1 (1 + t)β−1 1 Lsupercosh1 λ (1 + t)s dt B (α, β) −1 (c + 1)n α+β−1 = ·2 (6.288) n!

 β β+1 β+s−1 −n, s , s , · · · , s ; × 1+s Supercosh1+s λ2s , α+β α+β+1 α+β+s−1 c + 1, s , s , · · · , ; s where λ is a constant.

6.2 Extended works containing the Laguerre polynomials In this section, we present theorems for the special functions containing the Laguerre polynomials. Theorem 6.96 (Szegö theorem). [Szegö (1939)] Let α, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then S (α, z, t) = ez 0 F1 (−; α + 1; −zt) =

∞  k=0

where S (α, z, t) = ez

0 F1 (−; α

1 (α) L (t) zk , (1 + α)k k

+ 1; −zt) is called the Szegö function of first type.

(6.289)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

391

The result was obtained by Szegö in 1939 [159] and reported by Rainville in 1960 [45]. The generalized special case was reported by Chaundy in 1943 [186]. Theorem 6.97 (Rainville theorem). [Rainville (1960)] Let α, c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then T (α, c, z, t)

 zt = (1 − t) 1 F1 c; α + 1; − 1−t ∞ (α) k  (c)k Lk (z) t = , (1 + α)k −c

(6.290)

k=0

  zt is called the Rainville function. where T (α, c, z, t) = (1 − t)−c 1 F1 c; α + 1; − 1−t The result was obtained by Rainville in 1960 [45] and is the particular case reported by Burchnall and Chaundy in 1943 [186]. For more detail on the special functions, see Srivastava and Panda (1973) [187] and Carlitz (1961) [188]. Theorem 6.98 (Szegö theorem). [Szegö (1939)] We have ∞ 

zt

n −(1+α) − 1−t L(α) e , n (z) · t = (1 − t)

(6.291)

n=0

where t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. The result was obtained by Szegö in 1939 [159, p. 101] and rediscovered by Rainville in 1960 [45, p. 202]. The result is called the Szegö function of second type. Generally, we have that ∞ 

iλz

n −(1+α) − 1−z L(α) e , n (iλ) · z = (1 − z)

(6.292)

n=0 ∞ 

iλz

n −(1+α) 1−z L(α) e , n (−iλ) · z = (1 − z)

(6.293)

n=0

and ∞  n=0

λz

n −(1+α) 1−z L(α) e . n (−λ) · z = (1 − z)

(6.294)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.99. [Brafman (1957)] Let α, c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then [182]   zt zt + (1 + t)−c 1 F1 c; α + 1; (1 − t)−c 1 F1 c; α + 1; − 1−t 1+t   ⎡   zt 0 1 F1 c; α + 1; − 1−t 0 (1 − t)−c  ⎣ = −c 0 (1 + t) 0 1 F1 c; α + 1; =2

∞ (α)  (c)2k L (z) t 2k 2k

k=0

(1 + α)2k

⎤ zt 1+t

 ⎦

(6.295)

.

Theorem 6.100 (Brafman theorem). [Brafman (1957)] Let α, c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then ez 0 F1 (−; α + 1; −zt) + e−z 0 F1 (−; α + 1; zt)   z  0 0 e 0 F1 (−; α + 1; −zt) = 0 e−z 0 0 F1 (−; α + 1; zt) =2

∞ (α)  L (t) z2k 2k

k=0

(1 + α)2k

(6.296)

.

The result was obtained by Brafman in 1957 [182]. Theorem 6.101. Let α, c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then eiz 0 F1 (−; α + 1; −izt) + e−iz 0 F1 (−; α + 1; izt) = 2

∞ (α)  L (t) (−1)κ z2k 2k

(1 + α)2k

k=0

.

(6.297)

Proof. By means of the relations eiz 0 F1 (−; α + 1; −izt) =

∞ (α)  L (t) (iz)k k

(6.298)

(1 + α)k

k=0

and e−iz 0 F1 (−; α + 1; izt) =

∞ (α)  L (t) (−i)k zk k

k=0

(6.299)

(1 + α)k

we show that eiz 0 F1 (−; α + 1; −izt) + e−iz 0 F1 (−; α + 1; izt) = 2

∞ (α)  L (t) (−1)κ z2k 2k

k=0

(1 + α)2k

.

(6.300)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

393

Theorem 6.102. Let α, c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then eiz 0 F1 (−; α + 1; −izt) − e−iz 0 F1 (−; α + 1; izt) = 2i

(α) ∞  L2k+1 (t) (−1)k z2k+1

(1 + α)2k+1

k=0

.

(6.301)

Proof. By means of the relationships eiz 0 F1 (−; α + 1; −izt) =

∞ (α)  L (t) (iz)k k

(6.302)

(1 + α)k

k=0

and e−iz 0 F1 (−; α + 1; izt) =

∞ (α)  L (t) (−i)k zk k

(6.303)

(1 + α)k

k=0

we have eiz 0 F1 (−; α + 1; −izt) − e−iz 0 F1 (−; α + 1; izt) = 2i

(α) ∞  L2k+1 (t) (−1)k z2k+1

(1 + α)2k+1

k=0

.

(6.304)

Theorem 6.103. Let α, c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then ez 0 F1 (−; α + 1; −zt) − ez 0 F1 (−; α + 1; zt) = 2

(α) ∞  L2k+1 (t) z2k+1 k=0

(1 + α)2k+1

(6.305)

.

Proof. Using the relations eiz 0 F1 (−; α + 1; −izt) =

∞ (α)  L (t) (iz)k k

k=0

(6.306)

(1 + α)k

and e−iz 0 F1 (−; α + 1; izt) =

∞ (α)  L (t) (−1)k (iz)k k

k=0

(1 + α)k

(6.307)

,

we have ez 0 F1 (−; α + 1; −izt) − ez 0 F1 (−; α + 1; izt) = 2i

(α) ∞  L2k+ (t) (−1)k z2k+1 k=0

(1 + α)2k+1

.

(6.308)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.104. Let α, c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then   izt izt − (1 + it)−c 1 F1 c; α + 1; (1 − it)−c 1 F1 c; α + 1; − 1−t 1 + it = 2i

(α) ∞  (c)2k+1 L2k+1 (z) (−1)k t 2k+1

(1 + α)2k+1

k=0

(6.309)

.

Proof. By means of the relations (1 − it)

−c

 1 F1

izt c; α + 1; − 1−t

=

∞ (α)  (c)k L (z) (it)k k

(6.310)

(1 + α)k

k=0

and (1 + it)

−c

 1 F1

izt c; α + 1; 1 + it

=

∞ (α)  (c)k L (z) (−1)k (it)k k

(1 + α)k

k=0

(6.311)

we show that   izt izt − (1 + it)−c 1 F1 c; α + 1; (1 − it)−c 1 F1 c; α + 1; − 1−t 1 + it = 2i

(α) ∞  (c)2k+1 L2k+1 (t) (−1)k z2k+1

(1 + α)2k+1

k=0

(6.312)

.

Theorem 6.105. Let α, c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then −c

(1 − it) =2

 1 F1

izt c; α + 1; − 1−t

∞ (α)  (c)2k L (z) (−1)k t 2k 2k

k=0

(1 + α)2k

+ (1 + it)

−c

 1 F1

izt c; α + 1; 1 + it

(6.313)

.

Proof. Using the relations   ∞ (α) (c)k Lk (z) (it)k izt = (1 − it)−c 1 F1 c; α + 1; − 1−t (1 + α)k

(6.314)

  ∞ (α) (c)k Lk (z) (−1)k (it)k izt = , (1 + it)−c 1 F1 c; α + 1; 1 + it (1 + α)k

(6.315)

k=0

and

k=0

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

395

we have that

  izt izt + (1 + it)−c 1 F1 c; α + 1; (1 − it)−c 1 F1 c; α + 1; − 1−t 1 + it ∞ (α) k 2k  (c)2k L (t) (−1) z 2k =2 . (1 + α)2k

(6.316)

k=0

Theorem 6.106. Let α, c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then   zt zt −c −c − (1 + t) 1 F1 c; α + 1; (1 − t) 1 F1 c; α + 1; − 1−t 1+t =2

(α) ∞  (c)2k+1 L2k+1 (z) t 2k+1

(1 + α)2k+1

k=0

(6.317)

.

Proof. By the relations   ∞ (α) (c)k Lk (z) t k zt = (1 − t)−c 1 F1 c; α + 1; − 1−t (1 + α)k

(6.318)

  ∞ (α) (c)k Lk (z) (−1)k t k zt = (1 + t) 1 F1 c; α + 1; 1+t (1 + α)k

(6.319)

k=0

and

−c

k=0

we show that

  zt zt − (1 + t)−c 1 F1 c; α + 1; (1 − t)−c 1 F1 c; α + 1; − 1−t 1+t =2

(α) ∞  (c)2k+1 L2k+1 (z) t 2k+1

(1 + α)2k+1

k=0

(6.320)

.

Theorem 6.107. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then izt

zt

(1 − it)−(α+1) e 1−it − (1 + it)−(α+1) e− 1+it = 2i

∞ 

(α)

L2n+1 (z) (−1)n t 2n+1 .

(6.321)

n=0

Proof. In view of the relations izt

(1 − it)−(α+1) e 1−it =

∞  n=0

n L(α) n (z) (it)

(6.322)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and zt

(1 + it)−(α+1) e− 1+it =

∞ 

n n L(α) n (z) (−1) (it) ,

(6.323)

n=0

we obtain izt

zt

(1 − it)−(α+1) e 1−it − (1 + it)−(α+1) e− 1+it =

∞ 

n L(α) n (z) (it)

n=0 ∞ 

=2

−

∞ 

n n L(α) n (z) (−1) (it)

n=0

(6.324)

(α)

L2n+1 (z) (it)2n+1

n=0 ∞ 

= 2i

(α)

L2n+1 (z) (−1)n t 2n+1 .

n=0

Theorem 6.108. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then izt

zt

(1 − it)−(α+1) e 1−it + (1 + it)−(α+1) e− 1+it = 2

∞ 

(α)

L2n (z) (−1)n t 2n .

(6.325)

n=0

Proof. Due to the relations izt

(1 − it)−(α+1) e 1−it =

∞ 

n L(α) n (z) (it)

(6.326)

n=0

and zt

(1 + it)−(α+1) e− 1+it =

∞ 

n n L(α) n (z) (−1) (it) ,

(6.327)

n=0

we have izt

zt

(1 − it)−(α+1) e 1−it + (1 + it)−(α+1) e− 1+it =

∞ 

n L(α) n (z) (it) +

n=0 ∞ 

=2

∞ 

n n L(α) n (z) (−1) (it)

n=0 (α)

L2n (z) (it)2n

n=0

=2

∞  n=0

(α)

L2n (z) (−1)n t 2n .

(6.328)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

397

Theorem 6.109. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then zt

zt

(1 − t)−(α+1) e 1−t − (1 + t)−(α+1) e− 1+t = 2

∞ 

(α)

L2n+1 (z) t 2n+1 .

(6.329)

n=0

Proof. Using the relations zt

(1 − t)−(α+1) e 1−t =

∞ 

n L(α) n (z) t

(6.330)

n=0

and zt

(1 + t)−(α+1) e− 1+t =

∞ 

n L(α) n (z) (−t) ,

(6.331)

n=0

we have zt

zt

(1 − t)−(α+1) e 1−t − (1 + t)−(α+1) e− 1+t =

∞ 

n L(α) n (z) t −

n=0 ∞ 

=2

∞ 

n L(α) n (z) (−t)

(6.332)

n=0 (α)

L2n+1 (z) t 2n+1 .

n=0

Theorem 6.110. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then zt

zt

(1 − t)−(α+1) e 1−t + (1 + t)−(α+1) e− 1+t = 2

∞ 

(α)

L2n (z) t 2n .

(6.333)

n=0

Proof. By means of the relations zt

(1 − t)−(α+1) e 1−t =

∞ 

n L(α) n (z) t

(6.334)

n=0

and zt

(1 + t)−(α+1) e− 1+t =

∞ 

n L(α) n (z) (−t)

n=0

we show that zt

zt

(1 − t)−(α+1) e 1−t + (1 + t)−(α+1) e− 1+t

(6.335)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

=

∞ 

n L(α) n (z) t +

n=0 ∞ 

=2

∞ 

n L(α) n (z) (−t)

(6.336)

n=0 (α)

L2n (z) t 2n .

n=0

6.2.1 Hypergeometric supertrigonometric functions via the Rainville function In this section, we propose the hypergeometric supertrigonometric functions via Rainville function. Definition 6.17. The hypergeometric supersine via the Rainville function is defined as (α)

GLsupersink (t, z) =

∞ 

(α)

(−1)k L2k+1 (z) t 2k+1 ,

(6.337)

k=0

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.18. The hypergeometric supercosine via the Rainville function is defined as (α)

GLsupercos k (t, z) =

∞ 

(α)

(−1)k L2k (z) t 2k ,

(6.338)

k=0

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.19. The hypergeometric supertangent via the Rainville function is defined as (α)

GLsupertank (t, z) (α)

=

GLsupersink (t, z) (α)

GLsupercos k (t, z) ∞ k (α) 2k+1 k=0 (−1) L2k+1 (z) t , = ∞ k (α) 2k k=0 (−1) L2k (z) t

(6.339)

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.20. The hypergeometric supercotangent via the Rainville function is defined as GLsupercot (α) k (t, z) =

GLsupercos (α) k (t, z) (α)

GLsupersink (t, z)

(6.340)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions ∞

399

(α)

(−1)k L2k (z) t 2k = ∞ k=0 , k (α) 2k+1 k=0 (−1) L2k+1 (z) t where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.21. The hypergeometric supersecant via the Rainville function is defined as (α)

GLsupersec k (t, z) 1 = (α) GLsupercos k (t, z) 1 , = ∞ k (α) 2k k=0 (−1) L2k (z) t

(6.341)

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.22. The hypergeometric supercosecant via the Rainville function is defined as (α)

GLsupercosec k (t, z) 1 = (α) GLsupersink (t, z) 1 , = ∞ k (α) 2k+1 k=0 (−1) L2k+1 (z) t

(6.342)

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. (α)

The hypergeometric supersine GLsupersink (t, z), hypergeometric supercosine (α) (α) GLsupercos k (t, z), hypergeometric supertangent GLsupertank (t, z), hypergeometric su(α) (α) percotangent GLsupercot k (t, z), hypergeometric supersecant GLsupersec k (t, z), and (α) hypergeometric supercosecant GLsupercosec k (t, z) are called the hypergeometric supertrigonometric functions via the Rainville function. More generally, we have (α)

GLsupersink (λt, βz) =

∞ 

(α)

L2k+1 (βz) (−1)n (λt)2k+1 ,

(6.343)

k=0 (α)

GLsupercos k (λt, βz) =

∞  k=0

(α)

(−1)k L2k (βz) (λt)2k ,

(6.344)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

(α)

GLsupertank (λt, βz) (α)

=

GLsupersink (λt, βz) (α)

GLsupercos k (λt, βz) ∞ k (α) 2k+1 k=0 (−1) L2k+1 (βz) (λt) = ∞ , k (α) 2k k=0 (−1) L2k (βz) (λt)

(6.345)

GLsupercot (α) k (λt, βz) (α)

=

GLsupercos k (λt, βz) (α)

GLsupersink (λt, βz) ∞ k (α) 2k k=0 (−1) L2k (βz) (λt) = ∞ , k (α) 2k+1 k=0 (−1) L2k+1 (βz) (λt)

(6.346)

(α)

GLsupersec k (λt, βz) 1 = (α) GLsupercos k (λt, βz) 1 , = ∞ k (α) 2k k=0 (−1) L2k (βz) (λt)

(6.347)

and (α)

GLsupercosec k (λt, βz) 1 = (α) GLsupersink (λt, βz) 1 , = ∞ n (α) 2k+1 L (−1) (βz) (λt) k=0 2k+1

(6.348)

where λ ∈ R, α, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Thus we show that  izt izt 1  (α) (1 − it)−(α+1) e 1−it − (1 + it)−(α+1) e− 1+it = GLsupersink (t, z) 2i

(6.349)

 izt izt 1 (α) (1 − it)−(α+1) e 1−it + (1 + it)−(α+1) e− 1+it = GLsupercos k (t, z) . 2

(6.350)

and

Property 6.17. Let c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then izt

(α) (α) (1 − it)−(α+1) e 1−it = GLsupercoshk (t, z) + iGLsupersinhk (t, z)

(6.351)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

401

and izt

(1 + it)−(α+1) e− 1+it = GLsupercoshk (t, z) − iGLsupersinhk (t, z) . (α)

(α)

(6.352)

6.2.2 Hypergeometric superhyperbolic functions via the Rainville function We now consider the definitions of the hypergeometric superhyperbolic functions via Rainville function. Definition 6.23. The hypergeometric superhyperbolic supersine via the Rainville function is defined as ∞  (α) (α) GLsupersinhk (t, z) = L2k+1 (z) t 2k+1 , (6.353) k=0

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.24. The hypergeometric superhyperbolic supercosine via the Rainville function is defined as ∞  (α) (α) GLsupercoshk (t, z) = L2k (z) t 2k , (6.354) k=0

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.25. The hypergeometric superhyperbolic supertangent via the Rainville function is defined as (α)

GLsupertanhk (t, z) (α)

=

GLsupersinhk (t, z) (α)

GLsupercoshk (t, z) ∞ (α) (z) t 2k+1 k=0 L = ∞ 2k+1 , (α) 2k k=0 L2k (z) t

(6.355)

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.26. The hypergeometric superhyperbolic supercotangent via the Rainville function is defined as (α)

GLsupercothk (t, z) (α)

=

GLsupercoshk (t, z)

GLsupersinh(α) k (t, z) ∞ (α) L (z) t 2k , = ∞ k=0 (α)2k 2k+1 k=0 L2k+1 (z) t

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1.

(6.356)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Definition 6.27. The hypergeometric superhyperbolic supersecant via the Rainville function is defined as (α)

GLsupersechk (t, z) 1 = (α) GLsupercoshk (t, z) 1 , = ∞ (α) 2k k=0 L2k (z) t

(6.357)

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. The hypergeometric superhyperbolic supercosecant via the Rainville function is defined as (α)

GLsupercosec k (t, z) 1 = (α) GLsupersinhk (t, z) 1 = ∞ k (α) 2k+1 k=0 (−1) L2k+1 (z) t

(6.358)

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. (α)

Definition 6.28. The hypergeometric superhyperbolic supersine GLsupersinhk (t, z), hy(α) pergeometric superhyperbolic supercosine GLsupercoshk (t, z), hypergeometric superhy(α) perbolic supertangent GLsupertanhk (t, z), hypergeometric superhyperbolic supercotan(α) (α) gent GLsupercothk (t,z), hypergeometric superhyperbolic supersecant GLsupersechk (t,z), (α) and hypergeometric superhyperbolic supercosecant GLsupercosechk (t, z) are called the hypergeometric superhyperbolic functions via the Rainville function. More generally, we present (α)

GLsupersinhk (λt, βz) =

∞ 

(α)

L2k+1 (βz) (λt)2k+1 ,

(6.359)

k=0 (α) GLsupercoshk (λt, βz) =

∞ 

(α)

L2k (βz) (λt)2k ,

(6.360)

k=0 (α)

GLsupertanhk (λt, βz) (α)

=

GLsupersinhk (λt, βz) (α)

GLsupercoshk (λt, βz) ∞ (α) (βz) (λt)2k+1 k=0 L = ∞ 2k+1 , (α) 2k k=0 L2k (βz) (λt)

(6.361)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

403

(α)

GLsupercothk (λt, βz) (α)

=

GLsupercoshk (λt, βz) (α)

(6.362)

GLsupersinhk (λt, βz) ∞ (α) L (βz) (λt)2k , = ∞ k=0 (α)2k 2k+1 k=0 L2k+1 (βz) (λt) GLsupersech(α) k (λt, βz) 1 = (α) GLsupercoshk (λt, βz) 1 , = ∞ (α) 2k k=0 L2k (βz) (λt)

(6.363)

and (α)

GLsupercosechk (λt, βz) 1 = GLsupersinh(α) k (λt, βz) 1 , = ∞ (α) 2k+1 k=0 L2k+1 (βz) (λt)

(6.364)

where β, λ ∈ R, α, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. So, we show that

and

 zt zt 1 (α) (1 − t)−(α+1) e 1−t − (1 + t)−(α+1) e− 1+t = GLsupersinhk (t, z) 2

(6.365)

 zt zt 1 (α) (1 − t)−(α+1) e 1−t + (1 + t)−(α+1) e− 1+t = GLsupercoshk (t, z) . 2

(6.366)

Property 6.18. Let z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then zt

(1 − t)−(α+1) e 1−t = GLsupercoshk (t, z) + GLsupersinhk (t, z) (α)

(α)

(6.367)

and zt

(1 − t)−(α+1) e 1−t = GLsupercoshk (t, z) + GLsupersinhk (t, z) . (α)

(α)

(6.368)

6.2.3 Hypergeometric supertrigonometric functions via the Szegö function of first type We propose the hypergeometric supertrigonometric functions via the Szegö function of first type.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Definition 6.29. The hypergeometric supersine via the Szegö function of first type is defined as (α)

Ssupersink (t, z) 1 = (S (α, z, it) − S (α, z, −it)) 2i (α) ∞  (−1)κ L2k+1 (t) z2k+1 , = (1 + α)2k+1

(6.369)

k=0

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.30. The hypergeometric supercosine via the Szegö function of first type is defined as (α)

Ssupercos k (t, z) 1 = (S (α, z, it) − S (α, z, −it)) 2 ∞ (α)  (−1)κ L2k (t) z2k , = (1 + α)2k

(6.370)

k=0

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.31. The hypergeometric supertangent via the Szegö function of first type is defined as (α)

Ssupertank (t, z) (α)

=

=

Ssupersink (t, z) (α)

Ssupercos k (t, z) 2k+1 ∞ (−1)κ L(α) 2k+1 (t)z k=0

(1+α)2k+1

∞

(α)

(−1)κ L2k (t)z2k k=0 (1+α)2k

(6.371) ,

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.32. The hypergeometric supercotangent via the Szegö function of first type is defined as (α)

Ssupercot k (t, z) =

=

Ssupercos (α) k (t, z) (α)

Ssupersink (t, z) 2k ∞ (−1)κ L(α) 2k (t)z k=0

∞

(1+α)2k (α)

(−1)κ L2k+1 (t)z2k+1 k=0 (1+α)2k+1

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1.

(6.372) ,

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

405

Definition 6.33. The hypergeometric supersecant via the Szegö function of first type is defined as (α)

Ssupersec k (t, z) 1 = (α) Ssupercos k (t, z) 1 , = 2k ∞ (−1)κ L(α) 2k (t)z k=0

(6.373)

(1+α)2k

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.34. The hypergeometric supercosecant via the Szegö function of first type is defined as (α)

Ssupercosec k (t, z) 1 = (α) Ssupersink (t, z) 1 , = 2k+1 ∞ (−1)κ L(α) 2k+1 (t)z k=0

(6.374)

(1+α)2k+1

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. (α)

The hypergeometric supersine Ssupersink (t, z), hypergeometric supercosine (α) (α) Ssupercos k (t, z), hypergeometric supertangent Ssupertank (t, z), hypergeometric super(α) (α) cotangent Ssupercot k (t, z), hypergeometric supersecant Ssupersec k (t, z), and hyperge(α) ometric supercosecant Ssupercosec k (t, z) are called the hypergeometric supertrigonometric functions via the Szegö function of first type. More generally, we show that (α)

Ssupersink (βt, λz) =

(α) ∞  (−1)κ L2k+1 (βt) (λz)2k+1 k=0

(α)

Ssupercos k (βt, λz) =

(1 + α)2k+1

∞ (α)  (−1)κ L (βt) (λz)2k 2k

k=0

(1 + α)2k

,

,

(6.375)

(6.376)

(α)

Ssupertank (βt, λz) (α)

=

=

Ssupersink (βt, λz) (α)

Ssupercos k (βt, λz) 2k+1 ∞ (−1)κ L(α) 2k+1 (βt)(λz) k=0

∞

k=0

(1+α)2k+1 (α) κ (−1) L2k (βt)(λz)2k (1+α)2k

(6.377) ,

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

(α)

Ssupercot k (βt, λz) (α)

=

=

Ssupercos k (βt, λz) (α)

Ssupersink (βt, λz) 2k ∞ (−1)κ L(α) 2k (βt)(λz) k=0

(1+α)2k

∞

(α)

(−1)κ L2k+1 (βt)(λz)2k+1 k=0 (1+α)2k+1

(6.378) ,

(α)

Ssupersec k (βt, λz) 1 = (α) Ssupercos k (βt, λz) 1 , = 2k ∞ (−1)κ L(α) 2k (βt)(λz) k=0

(6.379)

(1+α)2k

and (α)

Ssupercosec k (βt, λz) 1 = (α) Ssupersink (βt, λz) 1 , = 2k+1 ∞ (−1)κ L(α) 2k+1 (βt)(λz) k=0

(6.380)

(1+α)2k+1

where β, λ ∈ R, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. In this case, we have the following result. Theorem 6.111. Let z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then 1 (α) (S (α, z, it) − S (α, z, −it)) = Ssupersink (t, z) 2i

(6.381)

1 (α) (S (α, z, it) + S (α, z, −it)) = Ssupercos k (t, z) . 2

(6.382)

and

Property 6.19. Let z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then (α)

(α)

S (α, z, it) = Ssupercos k (t, z) + iSsupersink (t, z)

(6.383)

(α) S (α, z, −it) = Ssupercos (α) k (t, z) − iSsupersink (t, z) .

(6.384)

and

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

407

6.2.4 Hypergeometric superhyperbolic functions via the Szegö function of first type We now consider the hypergeometric superhyperbolic functions via the Szegö function of first type. Definition 6.35. The hypergeometric superhyperbolic supersine via the Szegö function of first type is defined as (α)

Ssupersinhk (t, z) 1 = (S (α, z, t) − S (α, z, −t)) 2 (α) ∞  L2k+1 (t) z2k+1 = , (1 + α)2k+1

(6.385)

k=0

where c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.36. The hypergeometric superhyperbolic supercosine via Szegö function of first type is defined as (α)

Ssupercoshk (t, z) 1 = (S (α, z, t) + S (α, z, −t)) 2 ∞ (α)  L2k (t) z2k = , (1 + α)2k

(6.386)

k=0

where c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.37. The hypergeometric superhyperbolic supertangent via Szegö function of first type is defined as (α)

Ssupertanhk (t, z) (α)

=

=

Ssupersinhk (t, z) (α)

Ssupercoshk (t, z) 2k+1 ∞ L(α) 2k+1 (t)z k=0

∞

(1+α)2k+1 (α)

L2k (t)z2k k=0 (1+α)2k

where c, z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1.

,

(6.387)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Definition 6.38. The hypergeometric superhyperbolic supercotangent via Szegö function of first type is defined as Ssupercoth(α) k (t, z) (α)

=

=

Ssupercoshk (t, z) (α)

Ssupersinhk (t, z) 2k ∞ L(α) 2k (t)z k=0 (1+α)2k

∞

(α)

L2k+1 (t)z2k+1 k=0 (1+α)2k+1

(6.388)

,

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.39. The hypergeometric superhyperbolic supersecant via Szegö function of first type is defined as (α)

Ssupersechk (t, z) 1 = Ssupercosh(α) k (t, z) 1 , = 2k ∞ L(α) 2k (t)z

(6.389)

k=0 (1+α)2k

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.40. The hypergeometric superhyperbolic supercosecant via Szegö function of first type is defined as (α)

Ssupercosechk (t, z) 1 = (α) Ssupersinhk (t, z) 1 , = 2k+1 ∞ L(α) 2k+1 (t)z k=0

(6.390)

(1+α)2k+1

where z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. (α)

Definition 6.41. The hypergeometric superhyperbolic supersine Ssupersinh2k (t, z), hypergeometric superhyperbolic supercosine Ssupercosh(α) 2k (t, z), hypergeometric superhy(α) perbolic supertangent Ssupertanhk (t, z), hypergeometric superhyperbolic supercotan(α) gent Ssupercoth(α) k (t, z), hypergeometric superhyperbolic supersecant Ssupersechk (t, z), (α) and hypergeometric superhyperbolic supercosecant Ssupercosechk (t, z) are called the hypergeometric superhyperbolic functions via Szegö function of first type.

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

409

More generally, (α)

Ssupersinhk (βt, λz) =

(α) ∞  L2k+1 (βt) (λz)2k+1 k=0

(α)

Ssupercoshk (βt, λz) =

(1 + α)2k+1

∞ (α)  L (βt) (λz)2k 2k

k=0

(1 + α)2k

(α) Ssupertanhk (βt, λz) (α) Ssupersinhk (βt, λz) = (α) Ssupercoshk (βt, λz)

∞

=

,

,

(6.391)

(6.392)

(6.393)

(α)

L2k+1 (βt)(λz)2k+1 k=0 (1+α)2k+1

∞

(α)

L2k (βt)(λz)2k k=0 (1+α)2k

,

(α)

Ssupercothk (βt, λz) (α)

=

=

Ssupercoshk (βt, λz) (α)

Ssupersinhk (βt, λz) 2k ∞ L(α) 2k (βt)(λz) k=0

∞

k=0

(1+α)2k (α) L2k+1 (βt)(λz)2k+1 (1+α)2k+1

(6.394)

,

(α)

Ssupersechk (βt, λz) 1 = (α) Ssupercoshk (βt, λz) 1 , = 2k ∞ L(α) 2k (βt)(λz) k=0

(6.395)

(1+α)2k

and (α)

Ssupercosechk (βt, λz) 1 = (α) Ssupersinhk (βt, λz) 1 , = 2k+1 ∞ L(α) 2k+1 (βt)(λz) k=0

(1+α)2k+1

where β, λ ∈ R. In this case, we have the following results.

(6.396)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.112. Let z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then 1 (α) (S (α, z, t) − S (α, z, −t)) = Ssupersinhk (t, z) 2

(6.397)

1 (α) (S (α, z, t) + S (α, z, −t)) = Ssupercoshk (t, z) . 2

(6.398)

and

Property 6.20. Let z, t ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then (α)

(α)

(6.399)

(α)

(α)

(6.400)

S (α, z, t) = Ssupersinhk (t, z) + Ssupercoshk (t, z) and S (α, z, −t) = Ssupersinhk (t, z) − Ssupercoshk (t, z) .

6.2.5 Hypergeometric supertrigonometric functions via the Szegö function of second type We now consider the hypergeometric supertrigonometric functions via the Szegö function of second type. Definition 6.42. The hypergeometric supersine via the Szegö function of second type is defined as follows: (α)

GCsupersink (α, c, t, z)    1 izt izt −c −c = − (1 + it) 1 F1 c; α + 1; (1 − it) 1 F1 c; α + 1; − 2i 1 − it 1 + it =

(α) ∞  (−1)k (c)2k+1 L2k+1 (z) t 2k+1 k=0

(1 + α)2k+1

(6.401)

,

where t, c, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.43. The hypergeometric supercosine via the Szegö function of second type is defined as (α)

GCsupercos k (α, c, t, z)    1 izt izt = + (1 + it)−c 1 F1 c; α + 1; (1 − it)−c 1 F1 c; α + 1; − 2 1 − it 1 + it ∞ (α) k 2k  (−1) (c)2k L (z) t 2k = , (1 + α)2k k=0

where t, c, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1.

(6.402)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

411

Definition 6.44. The hypergeometric supertangent via the Szegö function of second type is defined as (α)

GCsupertank (α, c, t, z) (α)

=

=

GCsupersink (α, c, t, z) (α)

GCsupercos k (α, c, t, z) 2k+1 ∞ (−1)k (c)2k+1 L(α) 2k+1 (z)t k=0

(1+α)2k+1

∞

(α)

(−1)k (c)2k L2k (z)t 2k k=0 (1+α)2k

(6.403) ,

where t, c, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.45. The hypergeometric supercotangent via the Szegö function of second type is defined as (α)

GCsupercot k (α, c, t, z) (α)

=

=

GCsupercos k (α, c, t, z) (α)

GCsupersink (α, c, t, z) 2k ∞ (−1)k (c)2k L(α) 2k (z)t k=0

(1+α)2k

∞

(α)

(−1)k (c)2k+1 L2k+1 (z)t 2k+1 k=0 (1+α)2k+1

(6.404) ,

where t, c, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.46. The hypergeometric supersecant via the Szegö function of second type is defined as (α)

GCsupersec k (α, c, t, z) 1 = (α) GCsupercos k (α, c, t, z) 1 , = 2k ∞ (−1)k (c)2k L(α) 2k (z)t k=0

(6.405)

(1+α)2k

where t, c, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.47. The hypergeometric supercosecant via the Szegö function of second type is defined as GCsupercosec (α) k (α, c, t, z) 1 = (α) GCsupersink (α, c, t, z)

(6.406)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

=

1 ∞

(−1)k (c)

k=0

(α) 2k+1 2k+1 L2k+1 (z)t

,

(1+α)2k+1

where t, c, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. (α)

Definition 6.48. The hypergeometric supersine GCsupersink (α, c, t, z), hypergeometric (α) (α) supercosine GCsupercos k (α, c, t, z), hypergeometric supertangent GCsupertank (α,c,t,z), (α) hypergeometric supercotangent GCsupercot k (α, c, t, z), hypergeometric supersecant (α) (α) GCsupersec k (α, c, t, z), and hypergeometric supercosecant GCsupercosec k (α, c, t, z) are called the hypergeometric supertrigonometric functions via the Szegö function of second type. More generally, (α) GCsupersink (α, c, λt, βz) =

(α) ∞  (−1)k (c)2k+1 L2k+1 (βz) (λt)2k+1

(1 + α)2k+1

k=0 (α)

GCsupercos k (α, c, λt, βz) =

,

∞ (α)  (−1)k (c)2k L (βz) (λt)2k 2k

(1 + α)2k

k=0

,

(6.407)

(6.408)

(α)

GCsupertank (α, c, λt, βz) (α)

=

=

GCsupersink (α, c, λt, βz) (α)

GCsupercos k (α, c, λt, βz) 2k+1 ∞ (−1)k (c)2k+1 L(α) 2k+1 (βz)(λt) k=0

(1+α)2k+1

∞

(α)

(−1)k (c)2k L2k (βz)(λt)2k k=0 (1+α)2k

(6.409) ,

(α)

GCsupercot k (α, c, λt, βz) (α)

=

=

GCsupercos k (α, c, λt, βz) (α)

GCsupersink (α, c, λt, βz) 2k ∞ (−1)k (c)2k L(α) 2k (βz)(λt) k=0

(1+α)2k

∞

(α)

(−1)k (c)2k+1 L2k+1 (βz)(λt)2k+1 k=0 (1+α)2k+1

(6.410) ,

(α)

GCsupersec k (α, c, λt, βz) 1 = (α) GCsupercos k (α, c, λt, βz) 1 , = 2k ∞ (−1)k (c)2k L(α) 2k (βz)(λt) k=0

(1+α)2k

(6.411)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

413

and (α)

GCsupercosec k (α, c, λt, βz) 1 = (α) GCsupersink (α, c, λt, βz) 1 , = 2k+1 ∞ (−1)k (c)2k+1 L(α) 2k+1 (βz)(λt) k=0

(6.412)

(1+α)2k+1

where β, λ ∈ R, t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. In this case, we show the following results. Theorem 6.113. Let t, c, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then    1 izt izt −c −c − (1 + it) 1 F1 c; α + 1; (1 − it) 1 F1 c; α + 1; − 2i 1 − it 1 + it

(6.413)

(α)

= GCsupersink (α, c, t, z) and    izt izt 1 + (1 + it)−c 1 F1 c; α + 1; (1 − it)−c 1 F1 c; α + 1; − 2 1 − it 1 + it

(6.414)

(α) = GCsupercos k (α, c, t, z) .

Property 6.21. Let t, c, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then T (α, c, z, it) −c

= (1 − it)

 izt 1 F1 c; α + 1; − 1 − it (α)

(6.415) (α)

= GCsupercos k (α, c, t, z) + iGCsupersink (α, c, t, z) and T (α, c, z, −it)

 izt = (1 + it)−c 1 F1 c; α + 1; 1 − it

(6.416)

(α) = GCsupercos (α) k (α, c, t, z) − iGCsupersink (α, c, t, z) .

6.2.6 Hypergeometric superhyperbolic functions via the Szegö function of second type We now introduce the hypergeometric superhyperbolic functions via the Szegö function of second type.

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Definition 6.49. The hypergeometric superhyperbolic supersine via the Szegö function of second type is defined as (α)

GCsupersinhk (α, c, t, z)    1 zt zt = − (1 + t)−c 1 F1 c; α + 1; (1 − t)−c 1 F1 c; α + 1; − 2 1−t 1+t =

∞ 

(α) (c)2k+1 L2k+1 (z) t 2k+1

k=0

(1 + α)2k+1

(6.417)

,

where t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.50. The hypergeometric superhyperbolic supercosine via the Szegö function of second type is defined as (α)

GCsupercoshk (α, c, t, z)    1 zt zt = + (1 + t)−c 1 F1 c; α + 1; (1 − t)−c 1 F1 c; α + 1; − 2 1−t 1+t ∞ (α) 2k  (c)2k L (z) t 2k = , (1 + α)2k

(6.418)

k=0

where t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.51. The hypergeometric superhyperbolic supertangent via the Szegö function of second type is defined as (α)

GCsupertanhk (α, c, t, z) (α)

=

=

GCsupersinhk (α, c, t, z) GCsupercosh(α) k (α, c, t, z) (α) ∞ (c)2k+1 L2k+1 (z)t 2k+1 k=0

∞

(1+α)2k+1 (α)

(c)2k L2k (z)t 2k k=0 (1+α)2k

(6.419)

,

where t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.52. The hypergeometric superhyperbolic supercotangent via the Szegö function of second type is defined as GCsupercoth(α) k (α, c, t, z) =

GCsupercosh(α) k (α, c, t, z) (α)

GCsupersinhk (α, c, t, z)

(6.420)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

∞ =

k=0

∞

415

(α)

(c)2k L2k (z)t 2k (1+α)2k (α)

(c)2k+1 L2k+1 (z)t 2k+1 k=0 (1+α)2k+1

,

where t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.53. The hypergeometric superhyperbolic supersecant via the Szegö function of second type is defined as (α)

GCsupersechk (α, c, t, z) 1 = (α) GCsupercoshk (α, c, t, z) 1 , = 2k ∞ (c)2k L(α) 2k (z)t k=0

(6.421)

(1+α)2k

where t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Definition 6.54. The hypergeometric superhyperbolic supercosecant via the Szegö function of second type is defined as (α)

GCsupercosechk (α, c, t, z) 1 = (α) GCsupersinhk (α, c, t, z) 1 , = 2k+1 ∞ (c)2k+1 L(α) 2k+1 (z)t k=0

(6.422)

(1+α)2k+1

where t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. (α)

Definition 6.55. The hypergeometric supersine GCsupersinhk (α, c, t, z), hypergeometric (α) (α) supercosine GCsupercoshk (α, c, t, z), hypergeometric supertangent GCsupertanhk (α, c, (α) t, z), hypergeometric supercotangent GCsupercothk (α, c, t, z), hypergeometric superse(α) (α) cant GCsupersechk (α, c, t, z), and hypergeometric supercosecant GCsupercosechk (α, c, t, z) are called the hypergeometric superhyperbolic functions via the Szegö function of second type. More generally, (α) GCsupersinhk (α, c, λt, βz) =

(α) ∞  (c)2k+1 L2k+1 (βz) (λt)2k+1 k=0

(α)

GCsupercoshk (α, c, λt, βz) =

(1 + α)2k+1

∞ (α)  (c)2k L (βz) (λt)2k 2k

k=0

(1 + α)2k

,

,

(6.423)

(6.424)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

(α)

GCsupertanhk (α, c, λt, βz) (α)

=

=

GCsupersinhk (α, c, λt, βz) (α)

GCsupercoshk (α, c, λt, βz) 2k+1 ∞ (c)2k+1 L(α) 2k+1 (βz)(λt) k=0

∞

(1+α)2k+1 (α)

(c)2k L2k (βz)(λt)2k k=0 (1+α)2k

(6.425)

,

(α)

GCsupercothk (α, c, λt, βz) (α)

=

=

GCsupercoshk (α, c, λt, βz) (α)

GCsupersinhk (α, c, λt, βz) 2k ∞ (c)2k L(α) 2k (βz)(λt) k=0

∞

(1+α)2k (α)

(c)2k+1 L2k+1 (βz)(λt)2k+1 k=0 (1+α)2k+1

,

GCsupersech(α) k (α, c, λt, βz) 1 = (α) GCsupercoshk (α, c, λt, βz) 1 , = 2k ∞ (c)2k L(α) 2k (βz)(λt) k=0

(6.426)

(6.427)

(1+α)2k

and (α)

GCsupercosechk (α, c, λt, βz) 1 = (α) GCsupersinhk (α, c, λt, βz) 1 , = 2k+1 ∞ (c)2k+1 L(α) 2k+1 (βz)(λt) k=0

(6.428)

(1+α)2k+1

where β, λ ∈ R, t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. In this case, we have the following results. Theorem 6.114. Let t, c, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then we have    zt zt 1 −c −c − (1 + t) 1 F1 c; α + 1; (1 − t) 1 F1 c; α + 1; − 2 1−t 1+t (α) = GCsupersinhk (α, c, t, z)

(6.429)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

417

and    1 zt zt −c −c + (1 + t) 1 F1 c; α + 1; (1 − t) 1 F1 c; α + 1; − 2 1−t 1+t

(6.430)

(α) = GCsupercoshk (α, c, t, z) .

Theorem 6.115. Let t, c, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then we have T (α, c, z, t) = (1 − t)

−c

 1 F1

zt c; α + 1; − 1−t

(6.431)

(α)

(α)

= GCsupercoshk (α, c, t, z) + GCsupersinhk (α, c, t, z) and T (α, c, z, −t)

 zt = (1 + t)−c 1 F1 c; α + 1; 1+t

(6.432)

(α)

(α)

= GCsupercoshk (α, c, t, z) − GCsupersinhk (α, c, t, z) .

6.3 Some results based on the special functions In this section, we present the following results based on the new special functions. Theorem 6.116. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then et 0 Supersin1 (−; a + 1; −izt) =

∞ k  t 1 Lsupersin1 (−n; α + 1; z)

(1 + α)k

k=0

(6.433)

and e 1 Lsupercos 1 (−; a + 1; −izt) = t

∞ k  t 1 Lsupercos 1 (−n; α + 1; z)

(1 + α)k

k=0

.

(6.434)

Proof. Putting et 0 F1 (−; α + 1; −izt) =

∞ (α)  L (iz) t k k

(6.435)

(1 + α)k

k=0

and et 0 F1 (−; α + 1; izt) =

∞ (α)  L (−iz) t k k

k=0

(1 + α)k

,

(6.436)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

we have  et

1 (0 F1 (−; α + 1; −izt) − 0 F1 (−; α + 1; izt)) 2i

= −et 0 Supersin1 (−; a + 1; −izt)    (α) (α) 1 ∞ tk  2i Lk (iz) − Lk (−iz) = (1 + α)k k=0 ∞ k  t

=−

1 Lsupersin1 (−n; α

k=0

(6.437)

+ 1; λz)

(1 + α)k

and  e

t

1 (0 F1 (−; α + 1; −izt) + 0 F1 (−; α + 1; izt)) 2

= et 1 Lsupercos 1 (−; a + 1; −izt)    (α) (α) 1 ∞ tk L + L (iz) (−iz)  k k 2 = (1 + α)k

(6.438)

k=0

=

∞ k  t 1 Lsupercos 1 (−n; α + 1; z) k=0

(1 + α)k

,

where  1  (α) Ln (iz) − L(α) , (−iz) n 2i  1  (α) (α) L Lsupercos α + 1; z) = + L , (−n; (iz) (−iz) 1 1 n 2 n 1 (0 F1 (−; a + 1; iz) − 0 F1 (−; a + 1; −iz)) , 0 Supersin1 (−; a + 1; z) = 2i 1 Lsupersin1 (−n; α + 1; z) = −

(6.439) (6.440) (6.441)

and 0 Supercos 1 (−; a; z) =

1 (0 F1 (−; a + 1; iz) + 0 F1 (−; a + 1; −iz)) . 2

(6.442)

Theorem 6.117. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then et 0 Supersinh1 (−; a + 1; −zt) =

∞ k  t 1 Lsupersinh1 (−n; α + 1; z) k=0

(1 + α)k

(6.443)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

419

and et 1 Lsupercosh1 (−; a + 1; −zt) =

∞ k  t 1 Lsupercosh1 (−n; α + 1; z)

(1 + α)k

k=0

.

(6.444)

Proof. On writing et 0 F1 (−; α + 1; −zt) =

∞ (α)  L (z) t k k

(1 + α)k

k=0

,

(6.445)

,

(6.446)

and et 0 F1 (−; α + 1; zt) =

∞ (α)  L (−z) t k k

k=0

we have

 e

t

(1 + α)k

1 (0 F1 (−; α + 1; −zt) − 0 F1 (−; α + 1; zt)) 2



= −et 0 Supersinh1 (−; a + 1; −zt)    (α) (α) 1 ∞ tk  2 Lk (z) − Lk (−z) = (1 + α)k k=0 ∞ k  t

=−

1 Lsupersinh1 (−n; α

k=0

and

 e

t

(6.447)

+ 1; z)

(1 + α)k

1 (0 F1 (−; α + 1; −zt) + 0 F1 (−; α + 1; zt)) 2



= et 1 Lsupercosh1 (−; a + 1; −zt)    (α) (α) 1 ∞ tk  2 Lk (z) + Lk (−z) = (1 + α)k

(6.448)

k=0

=

∞ k  t 1 Lsupercosh1 (−n; α + 1; z) k=0

(1 + α)k

,

which can be derived from  1  (α) , Ln (z) − L(α) (−z) n 2  1  (α) Ln (z) + L(α) 1 Lsupercosh1 (−n; α + 1; z) = n (−z) , 2 1 0 Supersinh1 (−; a + 1; z) = (0 F1 (−; a + 1; z) − 0 F1 (−; a + 1; −z)) , 2 1 Lsupersinh1 (−n; α

+ 1; z) = −

(6.449) (6.450) (6.451)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and 0 Supercosh1 (−; a; z) =

1 (0 F1 (−; a + 1; z) + 0 F1 (−; a + 1; −z)) . 2

(6.452)

Theorem 6.118. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then −c

(1 − t)

 1 Supersin1

zt c; α + 1; 1−t

=

∞  (c)k t k 1 Lsupersin1 (−n; α + 1; z)

(1 + α)k

k=0

(6.453)

and   ∞ zt (c)k t k 1 Lsupercos 1 (−n; α + 1; z) = . (1 − t)−c 1 Supercos 1 c; α + 1; 1−t (1 + α)k

(6.454)

k=0

Proof. By the relations −c

(1 − t) and

 1 F1

izt c; α + 1; − 1−t

=

∞ (α)  (c)k L (iz) t k k

k=0

(1 + α)k

  ∞ (α) (c)k Lk (−iz) t k izt = (1 − t)−c 1 F1 c; α + 1; 1−t (1 + α)k

(6.455)

(6.456)

k=0

we get that 

  izt izt − 1 F1 c; α + 1; (1 − t) 1 F1 c; α + 1; − 1−t 1−t  zt = − (1 − t)−c 1 Supersin1 c; α + 1; 1−t    (α) (α) 1 ∞ (c) tk  k 2i Lk (iz) − Lk (−iz) = (1 + α)k −c

k=0 ∞ 

=−

k=0

1 2i



(c)k t k 1 Lsupersin1 (−n; α + 1; z) (1 + α)k

and     1 izt izt c; α + 1; − c; α + 1; + F F 1 1 1 1 2 1−t 1−t  zt = (1 − t)−c 1 Supercos 1 c; α + 1; 1−t

(1 − t)−c

(6.457)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

= =

∞ (c)  k k=0 ∞  k=0

  1 2

(α)

(α)

Lk (iz) + Lk (−iz)



421

tk (6.458)

(1 + α)k (c)k t k 1 Lsupercos 1 (−n; α + 1; z) , (1 + α)k

where  1  (α) , Ln (iz) − L(α) (−iz) n 2i  1  (α) Ln (iz) + L(α) , (−iz) 1 Lsupercos 1 (−n; α + 1; z) = n 2 1 (1 F1 (c; α + 1; iz) − 1 F1 (c; α + 1; −iz)) , 1 Supersin1 (c; α + 1; z) = 2i 1 Lsupersin1 (−n; α

+ 1; z) = −

(6.459) (6.460) (6.461)

and 1 Supercos 1 (c; α

+ 1; z) =

1 (1 F1 (c; α + 1; iz) + 1 F1 (c; α + 1; −iz)) . 2

(6.462)

Theorem 6.119. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then −c

(1 − t)

 1 Supersinh1

zt c; α + 1; 1−t

=

∞  (c)k t k 1 Lsupersinh1 (−n; α + 1; z)

(1 + α)k

k=0

(6.463)

and −c

(1 − t)

  ∞ zt (c)k t k 1 Lsupercosh1 (−n; α + 1; z) = . 1 Supercosh1 c; α + 1; 1−t (1 + α)k

(6.464)

k=0

Proof. By the relations   ∞ (α) (c)k Lk (z) t k zt = (1 − t) 1 F1 c; α + 1; − 1−t (1 + α)k

(6.465)

  ∞ (α) (c)k Lk (−z) t k zt = (1 − t)−c 1 F1 c; α + 1; 1−t (1 + α)k

(6.466)

−c

k=0

and

k=0

422

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

we obtain     1 zt zt − 1 F1 c; α + 1; (1 − t) 1 F1 c; α + 1; − 2 1−t 1−t  zt = − (1 − t)−c 1 Supersinh1 c; α + 1; 1−t    (α) (α) 1 ∞ (c) tk  k 2 Lk (z) − Lk (−z) = (1 + α)k −c

k=0 ∞ 

=−

k=0

(6.467)

(c)k t k 1 Lsupersinh1 (−n; α + 1; z) (1 + α)k

and     1 zt zt F F c; α + 1; − c; α + 1; + 1 1 1 1 2 1−t 1−t  zt = (1 − t)−c 1 Supercosh1 c; α + 1; 1−t    (α) (α) 1 ∞ (c) tk  k 2 Lk (z) + Lk (−z) = (1 + α)k

(1 − t)−c

(6.468)

k=0

=

∞  (c)k t k 1 Lsupercosh1 (−n; α + 1; z) k=0

(1 + α)k

,

from which we have  1  (α) Ln (z) − L(α) n (−z) , 2  1  (α) Ln (z) + L(α) 1 Lsupercosh1 (−n; α + 1; z) = n (−z) , 2 1 1 Supersinh1 (c; α + 1; z) = (1 F1 (c; α + 1; z) − 1 F1 (c; α + 1; −z)) , 2

(6.470)

1 (1 F1 (c; α + 1; z) + 1 F1 (c; α + 1; −z)) . 2

(6.472)

1 Lsupersinh1 (−n; α

+ 1; z) = −

(6.469)

(6.471)

and 1 Supercosh1 (c; α

+ 1; z) =

Theorem 6.120. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then (1 + t)α sin (zt) =

∞  n=0

t n 1 Lsupersin1 (−n; α − n + 1; z)

(6.473)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

423

and (1 + t)α cos (zt) =

∞ 

t n 1 Lsupercos 1 (−n; α − n + 1; z).

(6.474)

n=0

Proof. By means of the relations α −izt

(1 + t) e

=

∞ 

Ln(α−n) (iz) t n

(6.475)

Ln(α−n) (−iz) t n

(6.476)

n=0

and (1 + t)α eizt =

∞  n=0

we get  (1 + t)

α

 1  −izt izt e −e 2i

= − (1 + t)α sin (zt) ∞    1  (α−n) (α−n) Ln = (iz) − Ln (−iz) t n 2i n=0 ∞ 

=−

(6.477)

t n 1 Lsupersin1 (−n; α − n + 1; z)

n=0

and (1 + t)α

   1 −izt e + eizt 2

= (1 + t)α cos (zt) ∞     1 (α−n) Ln = (iz) + Ln(α−n) (−iz) t n 2 =

n=0 ∞ 

(6.478)

t n 1 Lsupercos 1 (−n; α − n + 1; z),

n=0

where 1 Lsupersin1 (−n; α

− n + 1; z) = −

 1  (α−n) Ln (iz) − Ln(α−n) (−iz) , 2i

and 1 Lsupercos 1 (−n; α

− n + 1; z) =

 1  (α−n) . Ln (iz) + L(α−n) (−iz) n 2

(6.479)

(6.480)

424

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.121. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then (1 + t)α sinh (zt) =

∞ 

t n 1 Lsupersinh1 (−n; α − n + 1; z)

(6.481)

t n 1 Lsupercosh1 (−n; α − n + 1; z).

(6.482)

n=0

and (1 + t)α cosh (zt) =

∞  n=0

Proof. By means of the relations (1 + t)α e−zt =

∞ 

L(α−n) (z) t n n

(6.483)

L(α−n) (−z) t n n

(6.484)

n=0

and (1 + t) e = α zt

∞  n=0

we get  (1 + t)α

 1  −zt e − ezt 2

= − (1 + t)α sinh (zt) ∞     1 (α−n) Ln = tn (z) − L(α−n) (−z) n 2 n=0 ∞ 

=−

(6.485)

t n 1 Lsupersinh1 (−n; α − n + 1; z)

n=0

and

 (1 + t)

α

 1  −zt zt e +e 2

= (1 + t)α cosh (zt) ∞     1 (α−n) (α−n) L = (z) + Ln (−z) t n 2 n

(6.486)

n=0

=

∞ 

t n 1 Lsupercosh1 (−n; α − n + 1; z),

n=0

where 1 Lsupersinh1 (−n; α

− n + 1; z) = −

 1  (α−n) Ln (z) − Ln(α−n) (−z) 2

(6.487)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

and 1 Lsupercosh1 (−n; α

− n + 1; z) =

 1  (α−n) Ln (z) + Ln(α−n) (−z) . 2

425

(6.488)

Theorem 6.122. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then 1 Lsupersin1 (−n; α

+ 1; z + t) =

n 

(β)

(6.489)

(β)

(6.490)

1 Lsupersin1 (−k; α

+ 1; z)Ln−k (t)

1 Lsupercos 1 (−k; α

+ 1; z)Ln−k (t).

k=0

and 1 Lsupercos 1 (−n; α

+ 1; z + t) =

n  k=0

Proof. By means of the relations L(α+β+1) (iz + t) = n

n 

(α)

(β)

Lk (iz) Ln−k (t)

(6.491)

k=0

and Ln(α+β+1) (−iz + t) =

n 

(α)

(β)

Lk (−iz) Ln−k (t)

(6.492)

k=0

we have + 1; z + t)   1 (α) L(α) =− n (iz + t) − Ln (−iz + t) 2i

 n n  1  (α) (β) (β) (α) =− Lk (iz) Ln−k (t) − Lk (−iz) Ln−k (t) 2i k=0 k=0 n    1  (α) (β) (α) Lk (iz) − Lk (−iz) Ln−k (t) − = 2i

1 Lsupersin1 (−n; α

k=0

=

n 

1 Lsupersin1 (−k; α

(β)

+ 1; z)Ln−k (t)

k=0

and + 1; z + t)   1 (α) Ln (iz + t) + L(α) = n (−iz + t) 2

1 Lsupercos 1 (−n; α

(6.493)

426

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

n n  1  (α) (β) (β) (α) Lk (iz) Ln−k (t) + Lk (−iz) Ln−k (t) = 2 k=0 k=0 n     1 (α) (β) (α) L (iz) + Lk (−iz) Ln−k (t) = 2 k

(6.494)

k=0

=

n 

1 Lsupercos 1 (−k; α

(β)

+ 1; z)Ln−k (t) ,

k=0

where  1  (α) Ln (iz) − L(α) n (−iz) , 2i  1  (α) Ln (iz) + L(α) 1 Lsupercos 1 (−n; α + 1; z) = n (−iz) , 2   1 (α) L(α) 1 Lsupersin1 (−n; α + 1; z + t) = − n (iz + t) − Ln (−iz + t) , 2i 1 Lsupersin1 (−n; α

+ 1; z) = −

(6.495) (6.496) (6.497)

and 1 Lsupercos 1 (−n; α

+ 1; z + t) =

 1  (α) Ln (iz + t) + L(α) n (−iz + t) . 2

(6.498)

Theorem 6.123. Let t, z ∈ C, n ∈ N, α > −1, |t| < 1, and |z| < 1. Then 1 Lsupersinh1 (−n; α

+ 1; z + t) =

n 

(β)

(6.499)

(β)

(6.500)

1 Lsupersinh1 (−k; α

+ 1; z)Ln−k (t)

1 Lsupercosh1 (−k; α

+ 1; z)Ln−k (t).

k=0

and 1 Lsupercosh1 (−n; α

+ 1; z + t) =

n  k=0

Proof. By means of the relations Ln(α+β+1) (z + t) =

n 

(β)

L(α) k (z) Ln−k (t)

(6.501)

k=0

and L(α+β+1) (−z + t) = n

n  k=0

(α)

(β)

Lk (−z) Ln−k (t)

(6.502)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

427

we have 1 Lsupersinh1 (−n; α

1

+ 1; z + t)

 (α) L(α) n (z + t) − Ln (−z + t) 2

 n n  1  (α) (β) (β) (α) Lk (z) Ln−k (t) − Lk (−z) Ln−k (t) =− 2 k=0 k=0 n    1  (α) (β) (α) − Lk (z) − Lk (−z) Ln−k (t) = 2 =−

(6.503)

k=0

=

n 

1 Lsupersinh1 (−k; α

(β)

+ 1; z)Ln−k (t)

k=0

and 1 Lsupercosh1 (−n; α

1

+ 1; z + t)

 (α) L(α) n (z + t) + Ln (−z + t) 2

 n n  1  (α) (β) (β) (α) Lk (z) Ln−k (t) + Lk (−z) Ln−k (t) = 2 k=0 k=0 n     1 (α) (β) (α) Lk (z) + Lk (−z) Ln−k (t) = 2 =

(6.504)

k=0

=

n 

1 Lsupercosh1 (−k; α

(β)

+ 1; z)Ln−k (t),

k=0

which follow from  1  (α) Ln (z) − L(α) n (−z) , 2  1  (α) Ln (z) + L(α) 1 Lsupercosh1 (−n; α + 1; z) = n (−z) , 2   1 (α) L(α) 1 Lsupersinh1 (−n; α + 1; z + t) = − n (z + t) − Ln (−z + t) , 2i 1 Lsupersinh1 (−n; α

+ 1; z) = −

(6.505) (6.506) (6.507)

and 1 Lsupercosh1 (−n; α

+ 1; z + t) =

 1  (α) Ln (z + t) + L(α) n (−z + t) . 2

(6.508)

428

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.124. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, z ∈ C, |z| < 1, and λ is a constant, then   z (c + 1)n B (c, d) c+d−1 −n, c; z t c−1 (z − t)d−1 L(α) F dt = λz . (6.509) (λt) 2 2 n α + 1, c + d; n! 0 Proof. By means of the relation (see Chapter 2) 



z

t

c−1

0

F1 (−n; α (z − t)d−1 1

+ 1; λt) dt = B (c, d) z

c+d−1

2 F2

−n, c; λz α + 1, c + d;

(6.510)

we can get 

z

t c−1 (z − t)d−1 F1 (−n; α + 1; λt) dt 1   z n! c−1 d−1 (α) t L (λt) dt = (z − t) (α + 1)n n 0  z n! = t c−1 (z − t)d−1 L(α) n (λt) dt (α + 1)n 0  −n, c; c+d−1 F2 = B (c, d) z2 λz , α + 1, c + d; 0

(6.511)

where Re (c) > 0, Re (d) > 0, |z| < 1, and λ is a constant. Theorem 6.125. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, z ∈ C, |z| < 1, and λ is a constant, then   z B (c, d) (α + 1)n c+d−1 −n, d; c−1 d−1 (α) z t Ln (λ (z − t)) dt = λz . (6.512) (z − t) 2 F2 α + 1, c + d; n! 0 Proof. Putting the relation 1 F1 (−n; α

+ 1; z) =

n! L(α) (z), (α + 1)n n

where n ∈ N, α > −1, and z ∈ C, we have  z t c−1 (z − t)d−1 1 F1 (−n; α + 1; λ (z − t)) dt 0  z n! t c−1 (z − t)d−1 L(α) (λ (z − t)) dt = (α + 1)n n 0  z n! t c−1 (z − t)d−1 L(α) = n (λ (z − t)) dt (α + 1)n 0  −n, d; c+d−1 λz , = B (c, d) z 2 F2 α + 1, c + d;

(6.513)

(6.514)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

429

which is derived from (see Chapter 2)   z −n, d; d−1 c−1 c+d−1 t λz , (6.515) (z − t)1 F1 (−n; α + 1; λ (z − t)) dt = B (c, d) z 2 F2 α + 1, c + d; 0 where Re (c) > 0, Re (d) > 0, |z| < 1, and λ is a constant. Theorem 6.126. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, z ∈ C, |z| < 1, and λ is a constant, then  z (α + 1)n B (c, d) c+d−1 z t c−1 (z − t)d−1 L(α) n (λt (z − t)) dt = n! 0

 (6.516) −n, c, d; λz2 × 3 F3 . c+d+1 4 α + 1, c+d 2 , 2 ; Proof. By the relation  z t c−1 (z − t)d−1 1 F1 (−n; α + 1; λt (z − t)) dt = B (c, d) zc+d−1 0

 −n, c, d; λz2 × 3 F3 c+d+1 4 α + 1, c+d 2 , 2 ; we can obtain



(6.517)

z

t c−1 (z − t)d−1 1 F1 (−n; α + 1; λt (z − t)) dt   z n! dt t c−1 (z − t)d−1 L(α) = − t)) (λt (z (α + 1)n n 0  z n! = t c−1 (z − t)d−1 L(α) n (λt (z − t)) dt (α + 1)n 0

 2 −n, c, d; λz = B (c, d) zc+d−1 3 F3 , c+d+1 4 α + 1, c+d 2 , 2 ; 0

which is derived from (see Chapter 2)  z t c−1 (z − t)d−1 1 F1 (−n; α + 1; λt (z − t)) dt 0

 −n, c, d; λz2 c+d−1 = B (c, d) z , 3 F3 c+d+1 4 α + 1, c+d 2 , 2 ; where Re (c) > 0, Re (d) > 0, |z| < 1, and λ is a constant. In this case, we have   1 (c + 1)n B (c, d) −n, c; c−1 d−1 (α) λ , t Ln (λt) dt = (1 − t) 2 F2 α + 1, c + d; n! 0

(6.518)

(6.519)

(6.520)

430

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions  0

1

t c−1 (1 − t)d−1 L(α) n (λ (1 − t)) dt =

 B (c, d) (α + 1)n −n, d; λ , F 2 2 α + 1, c + d; n!

(6.521)

and 

1

t c−1 (1 − t)d−1 L(α) n (λt (1 − t)) dt = 0

 −n, c, d; λ × 3 F3 , c+d+1 4 α + 1, c+d 2 , 2 ;

(α + 1)n B (c, d) n!

(6.522)

where Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, and λ is a constant. Moreover, 

1

−1

(1 + t)c−1 (1 − t)d−1 L(α) n (λ (1 + t)) dt = 

−n, c; 2λ , α + 1, c + d;

× 2 F2 

1

−1

(1 + t)c−1 (1 − t)d−1 L(α) n (λ (1 − t)) dt =

× 2 F2



−n, d; 2λ , α + 1, c + d;

(c + 1)n B (c, d) c+d−1 2 n!

B (c, d) (α + 1)n c+d−1 2 n!

(6.523)

(6.524)

and 

1

(1 + t)c−1 (1 − t)d−1 L(α) n (λt (1 − t)) dt = −1

 −n, c, d; × 3 F3 λ , c+d+1 α + 1, c+d 2 , 2 ;

(α + 1)n B (c, d) c+d−1 2 n!

(6.525)

where Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, and λ is a constant. Theorem 6.127. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, z ∈ C, |z| < 1, and λ is a constant, then  z n! t c−1 (z − t)d−1 1 Lsupersin1 (−n; α + 1; λt) dt = B (c, d) zc+d−1 (α + 1)n 0  (6.526) −n, c; λz , × 2 Supersin2 α + 1, c + d;  z n! t c−1 (z − t)d−1 1 Lsupercos 1 (−n; α + 1; λt) dt = B (c, d) zc+d−1 (α + 1)n 0  (6.527) −n, c; λz , × 2 Supercos 2 α + 1, c + d;

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions 

n! (α + 1)n

431

z

t c−1 (z − t)d−1 1 Lsupersinh1 (−n; α + 1; λt) dt = B (c, d) zc+d−1  −n, c; × 2 Supersinh2 λz , α + 1, c + d; 0

(6.528)

and 

n! (α + 1)n

z

t c−1 (z − t)d−1 1 Lsupercosh1 (−n; α + 1; λt) dt = B (c, d) zc+d−1  −n, c; × 2 Supercosh2 λz , α + 1, c + d; 0

(6.529)

where λ is a constant. Theorem 6.128. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, z ∈ C, |z| < 1, and λ is a constant, then n! (α + 1)n



z

t c−1 (z − t)d−1 1 Lsupersin1 (−n; α + 1; λt (z − t))dt = B (c, d) zc+d−1

 −n, c, d; λz2 × 3 Supersin3 , c+d+1 4 α + 1, c+d 2 , 2 ;  z n! t c−1 (z − t)d−1 1 Lsupercos 1 (−n; α + 1; λt (z − t))dt = B (c, d) zc+d−1 (α + 1)n 0

 −n, c, d; λz2 × 3 Supercos 3 , c+d+1 4 α + 1, c+d 2 , 2 ;  z n! t c−1 (z − t)d−1 1 Lsupersinh1 (−n; α + 1; λt (z − t))dt = B (c, d) zc+d−1 (α + 1)n 0

 −n, c, d; λz2 × 3 Supersinh3 , c+d+1 4 α + 1, c+d 2 , 2 ; 0

(6.530)

(6.531)

(6.532)

and n! (α + 1)n



z

t c−1 (z − t)d−1 1 Lsupercosh1 (−n; α + 1; λt (z − t))dt = B (c, d) zc+d−1

 −n, c, d; λz2 × 3 Supercosh3 , c+d+1 4 α + 1, c+d 2 , 2 ; 0

where λ is a constant.

(6.533)

432

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.129. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, z ∈ C, |z| < 1, and λ is a constant, then n! (α + 1)n



z

t c−1 (z − t)d−1 1 Lsupersin1 (−n; α + 1; λ (z − t)) dt = B (c, d) zc+d−1  −n, d; × 2 Supersin2 λz , α + 1, c + d;  z n! t c−1 (z − t)d−1 1 Lsupercos 1 (−n; α + 1; λ (z − t)) dt = B (c, d) zc+d−1 (α + 1)n 0  −n, d; × 2 Supercos 2 λz , α + 1, c + d;  z n! t c−1 (z − t)d−1 1 Lsupersinh1 (−n; α + 1; λ (z − t)) dt = B (c, d) zc+d−1 (α + 1)n 0  −n, d; × 2 Supersinh2 λz , α + 1, c + d; 0

(6.534)

(6.535)

(6.536)

and n! (α + 1)n



z

t c−1 (z − t)d−1 1 Lsupercosh1 (−n; α + 1; λ (z − t)) dt = B (c, d) zc+d−1  −n, d; × 2 Supercosh2 λz . α + 1, c + d; 0

(6.537)

Theorem 6.130. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, and λ is a constant, then n! (α + 1)n



1

t c−1 (1 − t)d−1 1 Lsupersin1 (−n; α + 1; λt) dt = B (c, d) 0  −n, c; × 2 Supersin2 λ , α + 1, c + d;  1 n! t c−1 (1 − t)d−1 1 Lsupercos 1 (−n; α + 1; λt) dt = B (c, d) (α + 1)n 0  −n, c; × 2 Supercos 2 λ , α + 1, c + d;  1 n! t c−1 (1 − t)d−1 1 Lsupersinh1 (−n; α + 1; λt) dt = B (c, d) (α + 1)n 0  −n, c; × 2 Supersinh2 λ , α + 1, c + d;

(6.538)

(6.539)

(6.540)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

433

and 

n! (α + 1)n

1

t c−1 (1 − t)d−1 1 Lsupercosh1 (−n; α + 1; λt) dt = B (c, d)  −n, c; λ . × 2 Supercosh2 α + 1, c + d; 0

(6.541)

Theorem 6.131. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, and λ is a constant, then n! (α + 1)n



1

t c−1 (1 − t)d−1 1 Lsupersin1 (−n; α + 1; λt (1 − t))dt = B (c, d)

 −n, c, d; λ × 3 Supersin3 , c+d+1 4 α + 1, c+d 2 , 2 ;  1 n! t c−1 (1 − t)d−1 1 Lsupercos 1 (−n; α + 1; λt (1 − t))dt = B (c, d) (α + 1)n 0

 −n, c, d; λ × 3 Supercos 3 , c+d+1 4 α + 1, c+d 2 , 2 ;  1 n! t c−1 (1 − t)d−1 1 Lsupersinh1 (−n; α + 1; λt (1 − t))dt = B (c, d) (α + 1)n 0

 −n, c, d; λ × 3 Supersinh3 , c+d+1 4 α + 1, c+d 2 , 2 ; 0

(6.542)

(6.543)

(6.544)

and n! (α + 1)n



1

t c−1 (1 − t)d−1 1 Lsupercosh1 (−n; α + 1; λt (1 − t))dt = B (c, d)

 −n, c, d; λ × 3 Supercosh3 . c+d+1 4 α + 1, c+d 2 , 2 ; 0

(6.545)

Theorem 6.132. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, and λ is a constant, then n! (α + 1)n



1

t c−1 (1 − t)d−1 1 Lsupersin1 (−n; α + 1; λ (1 − t)) dt = B (c, d)  −n, d; × 2 Supersin2 λ , α + 1, c + d;  1 n! t c−1 (1 − t)d−1 1 Lsupercos 1 (−n; α + 1; λ (1 − t)) dt = B (c, d) (α + 1)n 0  −n, d; λ , × 2 Supercos 2 α + 1, c + d; 0

(6.546)

(6.547)

434

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions 

n! (α + 1)n

1

t c−1 (1 − t)d−1 1 Lsupersinh1 (−n; α + 1; λ (1 − t)) dt = B (c, d) 0  −n, d; × 2 Supersinh2 λ , α + 1, c + d;

(6.548)

and 

n! (α + 1)n

1

t c−1 (1 − t)d−1 1 Lsupercosh1 (−n; α + 1; λ (1 − t)) dt = B (c, d)  −n, d; × 2 Supercosh2 λ , α + 1, c + d; 0

(6.549)

where λ is a constant. Theorem 6.133. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, and λ is a constant, then n! (α + 1)n



1

−1

(1 + t)c−1 (1 − t)d−1 1 Lsupersin1 (−n; α + 1; λt) dt = B (c, d) 2c+d−1 

× 2 Supersin2 n! (α + 1)n



1

−1



1

−1

(6.550)

(1 + t)c−1 (1 − t)d−1 1 Lsupercos 1 (−n; α + 1; λt) dt = B (c, d) 2c+d−1 

× 2 Supercos 2 n! (α + 1)n

−n, c; 2λ , α + 1, c + d;

−n, c; 2λ , α + 1, c + d;

(6.551)

(1 + t)c−1 (1 − t)d−1 1 Lsupersinh1 (−n; α + 1; λt) dt = B (c, d) 2c+d−1 

× 2 Supersinh2

−n, c; 2λ , α + 1, c + d;

(6.552)

and n! (α + 1)n



1

−1

(1 + t)c−1 (1 − t)d−1 1 Lsupercosh1 (−n; α + 1; λt) dt = B (c, d) 2c+d−1

× 2 Supercosh2



−n, c; 2λ , α + 1, c + d;

where λ is a constant.

(6.553)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

435

Theorem 6.134. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, z ∈ C, |z| < 1, and λ is a constant, then n! (α + 1)n



1

(1 + t)c−1 (1 − t)d−1 1 Lsupersin1 (−n; α + 1; λt (1 − t)) dt

−1 c+d−1

= B (c, d) 2



× 3 Supersin3 n! (α + 1)n



1

c+d+1 α + 1, c+d 2 , 2 ;

λ ,

(1 + t)c−1 (1 − t)d−1 1 Lsupercos 1 (−n; α + 1; λt (1 − t)) dt

−1 c+d−1

= B (c, d) 2



× 3 Supercos 3 n! (α + 1)n

−n, c, d;

(6.554)





1

−1

−n, c, d;

c+d+1 α + 1, c+d 2 , 2 ;

(6.555)

 λ ,

(1 + t)c−1 (1 − t)d−1 1 Lsupersinh1 (−n; α + 1; λt (1 − t)) dt

= B (c, d)



× 2c+d−1 3 Supersinh3

−n, c, d; c+d+1 α + 1, c+d 2 , 2 ;



(6.556)

λ ,

and n! (α + 1)n



1

−1

(1 + t)c−1 (1 − t)d−1 1 Lsupercosh1 (−n; α + 1; λt (1 − t)) dt

= B (c, d)



× 2c+d−1 3 Supercosh3

−n, c, d;

c+d+1 α + 1, c+d 2 , 2 ;



(6.557)

λ .

Theorem 6.135. If Re (c) > 0, Re (d) > 0, n ∈ N, α > −1, and λ is a constant, then n! (α + 1)n



1

−1

(1 + t)c−1 (1 − t)d−1 1 Lsupersin1 (−n; α + 1; λ (1 − t)) dt = B (c, d) 

× 2c+d−1 2 Supersin2 n! (α + 1)n ×2



c+d−1

1 −1

−n, d; 2λ , α + 1, c + d;

(6.558)

(1 + t)c−1 (1 − t)d−1 1 Lsupercos 1 (−n; α + 1; λ (1 − t)) dt = B (c, d)

2 Supercos 2



−n, d; 2λ , α + 1, c + d;

(6.559)

436

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

n! (α + 1)n ×2



1 −1

c+d−1

(1 + t)c−1 (1 − t)d−1 1 Lsupersinh1 (−n; α + 1; λ (1 − t)) dt = B (c, d) 

−n, d; 2λ , α + 1, c + d;

2 Supersinh2

(6.560)

and n! (α + 1)n



1

−1

(1 + t)c−1 (1 − t)d−1 1 Lsupercosh1 (−n; α + 1; λ (1 − t)) dt = B (c, d) 

× 2c+d−1 2 Supercosh2

−n, d; 2λ , α + 1, c + d;

(6.561)

where λ is a constant. Theorem 6.136 (Laplace transform). Let α, β, z, λ ∈ C. Then the Laplace transform of t λ−1 S (α, z, βt) is as follows: 

βz λ, λ−1 z  (λ) , S (α, z, βt) = e · λ · 1 F1 L t ;− α+1 s s

(6.562)

where the Laplace transform of a function f is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(6.563)

0

In this case, it is not difficult to show the following theorems. Theorem 6.137. If α, β, ϑ, z ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then  z   1 t α−1 (z − t)β−1 S c, ϑ, λt κ dt = ez · zα+β−1 B (α, β) 0

 (6.564) α α+κ−1 ; κ ,··· , κ κ λϑz × κ F1+κ , α+β+κ−1 ; c + 1, α+β κ ,··· , κ where λ ∈ C is a constant. Theorem 6.138. If α, β, ϑ, z ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 S c, ϑ, λt κ (z − t)s dt B (α, β) 0 = ez · zα+β−1 × κ+s F1+κ+s where λ ∈ C is a constant.

α

α+κ−1 β , s , · · · , β+s−1 ; κ ,··· , κ s α+β α+β+κ+s−1 c + 1, κ+s , · · · , ; κ+s

κ κ s s λϑzκ+s (κ + s)κ+s

(6.565)

 ,

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

437

Theorem 6.139. If α, β, ϑ, z ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have 

  t α−1 (z − t)β−1 S c, ϑ, λ (z − t)s dt = ez · zα+β−1

0  β β+1 β+s−1 ; s , s ,··· , s s × s F1+s λϑz , α+β+1 , · · · , α+β+s−1 ; c + 1, α+β s , s s

1 B (α, β)

z

(6.566)

where λ ∈ C is a constant. Theorem 6.140. If α, β, ϑ ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then 

  t α−1 (1 − t)β−1 S c, ϑ, λt κ dt = e 

0 α α+κ−1 ; κ ,··· , κ × κ F1+κ λϑ , α+β+κ−1 ; c + 1, α+β κ ,··· , κ

1 B (α, β)

1

(6.567)

where λ ∈ C is a constant. Theorem 6.141. If α, β ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have 1 B (α, β)



1

  t α−1 (1 − t)β−1 S c, ϑ, λt κ (1 − t)s dt

0

=e

α

α+κ−1 β , s , · · · , β+s−1 ; κ κ s s λϑ κ ,··· , κ s α+β α+β+κ+s−1 (κ + s)κ+s ; c + 1, κ+s , · · · , κ+s

× κ+s F1+κ+s

(6.568)

 ,

where λ ∈ C is a constant. Theorem 6.142. If α, β, ϑ ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have 

  t α−1 (1 − t)β−1 S c, ϑ, λ (1 − t)s dt = e 

0 β β+1 β+s−1 ; s , s ,··· , s × s F1+s λϑ , α+β+1 α+β+s−1 c + 1, α+β , , · · · , ; s s s

1 B (α, β)

where λ ∈ C is a constant.

1

(6.569)

438

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.143. If α, β, ϑ ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then  1   1 (1 − t)α−1 (1 + t)β−1 S c, ϑ, λ (1 − t)κ dt = e2 · 2α+β−1 B (α, β) −1

 (6.570) α α+κ−1 ; κ ,··· , κ κ × κ F1+κ , α+β+κ−1 λϑ2 ; c + 1, α+β κ ,··· , κ where λ ∈ C is a constant. Theorem 6.144. If α, β, ϑ ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have 1 B (α, β) =e ·2 2



1

  (1 − t)α−1 (1 + t)β−1 S c, ϑ, λ (1 − t)κ (1 + t)s dt

−1 α+β−1

× κ+s F1+κ+s

α

α+κ−1 β , s , · · · , β+s−1 ; κ ,··· , κ s α+β α+β+κ+s−1 ; c + 1, κ+s , · · · , κ+s

κ κ s s λϑ2κ+s (κ + s)κ+s

(6.571)

 ,

where λ ∈ C is a constant. Theorem 6.145. If α, β, ϑ ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 (1 − t)α−1 (1 + t)β−1 S c, ϑ, λ (1 + t)s dt = e2 · 2α+β−1 B (α, β) −1

 (6.572) β β+1 β+s−1 ; s , s ,··· , s s × s F1+s λϑ2 , α+β+1 c + 1, α+β , · · · , α+β+s−1 ; s , s s where λ ∈ C is a constant. Theorem 6.146. Let α, β, z, λ ∈ C,  (λ) = 0, and n ∈ N0 . Then  e · 1 F1 z

  ∞ k z λ, λ, −k · 2 F1 ; −βz = ;β . α+1 α+1 k! k=0

Proof. We have  0

∞

 1 (α) k e e L (βt) z dt, t (1 + α)k k 0 k=0   ∞   λ, −t λ z z ; −βz , e t e 0 F1 (−; α + 1; −βzt) dt = e ·  (λ) · 1 F1 α+1 0 −t

λ z  t e 0 F1 (−; α + 1; −βzt) dt =



∞



−t

λ

∞ 

(6.573)

(6.574)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions 



∞

e

−t

0

t

λ

∞  k=0

  ∞ k  1 z λ, −k (α) k · 2 F1 L (βt) z dt =  (λ) · ;β , α+1 k! (1 + α)k k

(6.575)

k=0

so that  (λ) ·

439

∞ k  z k=0

k!

· 2 F1

  λ, −k λ, ; β = ez ·  (λ) · 1 F1 ; −βz , α+1 α+1

(6.576)

which is derived from   λ, −k λ, z · 2 F1 ; β = e 1 F1 ; −βz , α+1 α+1 k!

∞ k  z k=0

(6.577)

where  (λ) = 0. Theorem 6.147. Let α, β, z, λ ∈ C,  (λ) = 0, and n ∈ N0 . Then: (I) e

(β−1)z

=

∞ k  z k=0

k!

 · 2 F1

α + 1, −k ;β ; α+1

(6.578)

(II)  α + 1, −k ; 1 = 1; α+1

(6.579)

 ∞  1 α + 1, −k · 2 F1 ; 1 = 1. α+1 k!

(6.580)

∞ k  z k=0

k!

· 2 F1

(III)

k=0

Proof. Taking λ = α + 1, we have  1 F1

α+1 ; −βz = e−βz , α+1

so that e(β−1)z =

∞ k  z k=0

k!

 · 2 F1

α + 1, −k ;β . α+1

(6.581)

On setting β = 1, we get ∞ k  z k=0

k!

 · 2 F1

α + 1, −k ; 1 = 1. α+1

(6.582)

440

An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Taking z = 1, we get

 ∞  1 α + 1, −k · 2 F1 ; 1 = 1. α+1 k!

(6.583)

k=0

Theorem 6.148 (Laplace transform). Let α, β, z, λ ∈ C. Then the Laplace transform of t λ−1 S (α, z, βt) is as follows: 

βz λ, λ−1 z  (λ) , S (α, z, βt) = e · λ · 1 F1 L t ;− α+1 s s

(6.584)

where the Laplace transform of a function f is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(6.585)

0

Based on the previous results, we can derive the following theorems. Theorem 6.149. If α, β, ϑ, z ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then  z   1 t α−1 (z − t)β−1 S c, ϑ, λt κ dt = ez · zα+β−1 B (α, β) 0

 (6.586) α α+κ−1 ; κ ,··· , κ κ λϑz × κ F1+κ , α+β+κ−1 c + 1, α+β ; κ ,··· , κ where λ ∈ C is a constant. Theorem 6.150. If α, β, ϑ, z ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have  z   1 t α−1 (z − t)β−1 S c, ϑ, λt κ (z − t)s dt B (α, β) 0 = ez · zα+β−1 × κ+s F1+κ+s

α

α+κ−1 β , s , · · · , β+s−1 ; κ κ s s λϑzκ+s κ ,··· , κ s κ+s α+β α+β+κ+s−1 ; (κ + s) c + 1, κ+s , · · · , κ+s

(6.587)

 ,

where λ ∈ C is a constant. Theorem 6.151. If α, β, ϑ, z ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  z   1 t α−1 (z − t)β−1 S c, ϑ, λ (z − t)s dt = ez · zα+β−1 B (α, β) 0

 (6.588) β β+1 β+s−1 ; s , s ,··· , s s × s F1+s λϑz , α+β+1 c + 1, α+β , · · · , α+β+s−1 ; s , s s where λ ∈ C is a constant.

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions

441

Theorem 6.152. If α, β, ϑ ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then 

  t α−1 (1 − t)β−1 S c, ϑ, λt κ dt = e 

0 α α+κ−1 ; κ ,··· , κ × κ F1+κ λϑ , α+β+κ−1 c + 1, α+β ; κ ,··· , κ

1 B (α, β)

1

(6.589)

where λ ∈ C is a constant. Theorem 6.153. If α, β ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have 1 B (α, β)



1

  t α−1 (1 − t)β−1 S c, ϑ, λt κ (1 − t)s dt

0

=e

α

α+κ−1 β , s , · · · , β+s−1 ; κ ,··· , κ s α+β α+β+κ+s−1 c + 1, κ+s , · · · , ; κ+s

× κ+s F1+κ+s

κ κ s s λϑ (κ + s)κ+s

(6.590)

 ,

where λ ∈ C is a constant. Theorem 6.154. If α, β, ϑ ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have 

  t α−1 (1 − t)β−1 S c, ϑ, λ (1 − t)s dt = e 

0 β β+1 β+s−1 , , · · · , ; s s s × s F1+s λϑ , α+β+1 , , · · · , α+β+s−1 ; c + 1, α+β s s s

1 B (α, β)

1

(6.591)

where λ ∈ C is a constant. Theorem 6.155. If α, β, ϑ ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ ∈ N, then 

  (1 − t)α−1 (1 + t)β−1 S c, ϑ, λ (1 − t)κ dt = e2 · 2α+β−1 −1

 α α+κ−1 ; κ ,··· , κ κ × κ F1+κ , α+β+κ−1 λϑ2 ; c + 1, α+β κ ,··· , κ

1 B (α, β)

1

where λ ∈ C is a constant.

(6.592)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 6.156. If α, β, ϑ ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and κ, s ∈ N, then we have 1 B (α, β)



1

−1

  (1 − t)α−1 (1 + t)β−1 S c, ϑ, λ (1 − t)κ (1 + t)s dt

= e2 · 2α+β−1

α × κ+s F1+κ+s

α+κ−1 β , s , · · · , β+s−1 ; κ ,··· , κ s α+β α+β+κ+s−1 ; c + 1, κ+s , · · · , κ+s

κ κ s s λϑ2κ+s (κ + s)κ+s

(6.593)

 ,

where λ ∈ C is a constant. Theorem 6.157. If α, β, ϑ ∈ C, Re (α) > 0, Re (β) > 0, Re (c) > 0, Re (c + 1) > −n, and s ∈ N, then we have  1   1 (1 − t)α−1 (1 + t)β−1 S c, ϑ, λ (1 + t)s dt = e2 · 2α+β−1 B (α, β) −1

 (6.594) β β+1 β+s−1 ; s , s ,··· , s s λϑ2 , × s F1+s α+β+1 c + 1, α+β , · · · , α+β+s−1 ; s , s s where λ ∈ C is a constant. Theorem 6.158. Let α, β, z, λ ∈ C,  (λ) = 0, and n ∈ N0 . Then  e · 1 F1 z

  ∞ k z λ, λ, −k · 2 F1 ; −βz = ;β . α+1 α+1 k! k=0

Proof. Putting the relations  0

 1 (α) k e e L (βt) z dt, t (1 + α)k k 0 k=0   ∞   λ, ; −βz , e−t t λ ez 0 F1 (−; α + 1; −βzt) dt = ez ·  (λ) · 1 F1 α+1 0

   ∞ ∞ ∞ k   1 z λ, −k (α) −t λ k ;β , · 2 F1 e L (βt) z dt =  (λ) · t α+1 k! (1 + α)k k 0

∞

−t

λ z  t e 0 F1 (−; α + 1; −βzt) dt =

k=0



∞



−t

λ

∞ 

(6.595)

(6.596)

(6.597)

k=0

we have  (λ) ·

  λ, −k λ, z ; β = e ·  (λ) · 1 F1 ; −βz , · 2 F1 α+1 α+1 k!

∞ k  z k=0

which leads to the representation

(6.598)

Chapter 6 • Hypergeometric supertrigonometric and superhyperbolic functions   λ, −k λ, z ; β = e 1 F1 ; −βz , · 2 F1 α+1 α+1 k!

∞ k  z k=0

443

(6.599)

where  (λ) = 0. Theorem 6.159. Let α, β, z, λ ∈ C,  (λ) = 0, and n ∈ N0 . Then: (I) e(β−1)z =

∞ k  z k=0

(II)

k!

 · 2 F1

α + 1, −k ;β ; α+1

 α + 1, −k ; 1 = 1; · 2 F1 α+1 k!

(6.601)

 ∞  1 α + 1, −k · 2 F1 ; 1 = 1. α+1 k!

(6.602)

∞ k  z k=0

(III)

(6.600)

k=0

Proof. Taking λ = α + 1, we have  α + 1, ; −βz = e−βz , 1 F1 α+1 so that e

(β−1)z

=

∞ k  z k=0

k!

Taking β = 1, we have

· 2 F1



α + 1, −k ;β . α+1

(6.603)

α + 1, −k ; 1 = 1. α+1

(6.604)

 ∞  1 α + 1, −k ; 1 = 1. · 2 F1 α+1 k!

(6.605)

∞ k  z k=0

Taking z = 1, we get



k!

· 2 F1

k=0

For more detail of the Laguerre polynomials, see Szegö (1948) [189], Turán (1950) [190], Courant and Hilbert (1953) [191], Jackson (1941) [192], Borwein et al. (1994) [193], Sun (2011) [194], Ram et al. (2010) [195], McCarthy et al. (1993) [196], Mukherjee and Acton (2014) [197], Szegö (1936) [169], Everitt et al. (2002) [198], Boyd (2004) [199], Dattoli et al. (2001) [200], Alpert and Rokhlin (1991) [201], Neves et al. (2006) [202], Sogo (1996) [203],

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Paton (1975) [204], Beukers (1980) [205], Rad et al. (2014) [206], Tohidi and Samadi (2012) [207], Tien (2009) [208], Wan and Zudilin (2013) [209], Singh et al. (2010) [210], Dougall (1953) [211], Wünsche (1998) [212], Watson (1933) [213], Whipple (1925) [214], Darling (1935) [215], Fasenmyer (1947) [216], Fields and Wimp (1961) [217], Srivastava (1982) [218], Weisner (1959) [219], Beals and Wong (2016) [220], Burchnall and Chaundy (1948) [221], Srivastava (1969) [222], Srivastava and Buschman (1975) [223], Srivastava (1969) [224], Abdul-Halim and Al-Salam (1963) [225], Burchnall and Chaundy (1940) [226], Srivastava (1971) [227], and Burchnall and Chaundy (1941) [228].

7 Hypergeometric supertrigonometric and superhyperbolic functions via Legendre polynomials 7.1 Legendre polynomials In this section, we introduce the definition, properties, and theorems for the Legendre polynomials.

7.1.1 Definition, properties, and theorems for the Legendre polynomials We now begin with the definition of the Legendre polynomials. Definition 7.1. [Legendre (1785)] The Legendre polynomial of degree n, denoted by Pn (z), is defined as [229] Pn (z) =

n  (n!)2 (z − 1)k (z + 1)n−k k=0

2n k! (n − k)! (1)k (1)n−k

,

(7.1)

where −1 < z < 1 and n ∈ N0 . The result was discovered by Legendre in 1785 [229] and reported by Bhonsle in 1957 [230]. Property 7.1. [Rodrigues (1816)] If −1 < z < 1 and n ∈ N0 , then [171] Pn (z) =

n 1 d2  2 z − 1 . 2n n! dz2

(7.2)

This result is known as the Rodrigues formula discovered by Rodrigues in 1816 [171], reported by Whittaker in 1902 [18], and discussed by Whittaker and Watson in 1927 [19]. Property 7.2. [Rainville (1960)] If −1 < z < 1 and n ∈ N0 , then [28]   1 n   2 n (2z) n n 1 1 1 − . , − + ; − n; Pn (z) = F 2 1 n! 2 2 2 2 z2 An Introduction to Hypergeometric, Supertrigonometric, and Superhyperbolic Functions https://doi.org/10.1016/B978-0-12-824154-7.00013-1 Copyright © 2021 Elsevier Inc. All rights reserved.

(7.3) 445

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The result was obtained by Rainville in 1960 [28]. Property 7.3. [Glaisher theorem] If −1 < z < 1 and n ∈ N0 , then [19]   ∞   d n 1 2 2 2 2 − Pn (z) = √ e− 1−z t e−z t dt. dz n! π −∞

(7.4)

The result was obtained by Glaisher and indirectly reported by Whittaker and Watson in 1927 (see [19], p. 332). Based on the Glaisher theorem, we derive an alternative form: If −1 < z < 1 and n ∈ N0 , then [19]   ∞   d n 2 2 2 2 2 − (7.5) Pn (z) = √ e− 1−z t e−z t dt. dz n! π 0 Theorem 7.1 (Schläfli’s integral formula). [Schläfli (1881)] If −1 < z < 1 and n ∈ N0 , then [231] 2 n  t −1 1 Pn (z) = dt, 2πi C 2n (t − z)n+1

(7.6)

where C is a contour that encircles the point z once counterclockwise. This result is known as the Schläfli’s integral formula discovered by Schläfli in 1881 [231] and reported by Whittaker and Watson in 1927 [19]. Theorem 7.2 (Whittaker theorem). [Whittaker (1902)] If −1 < z < 1 and n ∈ N0 , then the function u (z) = Pn (z) is the solution of the ordinary differential equation [18] 

1 − z2

 ∂ 2u ∂z2

− 2z

∂u + n (n + 1) = 0, ∂z

(7.7)

where −1 < z < 1 and n ∈ N0 . The result was obtained by Whittaker in 1902 [18]. Property 7.4. [Legendre (1785, 1789)] If −1 < z < 1 and m, n ∈ N0 , then 

1

−1

and



Pn (t) Pn (t) dt =

1

−1

where m, n ∈ N0 , m = n.

2 2n + 1

Pm (t) Pn (t) dt = 0,

(7.8)

(7.9)

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

447

These integral properties of the Legendre polynomials were obtained by Legendre [229, 232] and reported by Whittaker in 1902 [18]. Theorem 7.3 (Clare theorem). [Clare (1908)] If −1 < z < 1 and n ∈ N0 , then [19] 

  1 1 1 −1 −(2n+1)z e . (cosh (2z) − t) 2 Pn (t) dt = 2 2 n + 2 −1 1

(7.10)

The result was obtained by Clare in 1908 and indirectly reported by Whittaker and Watson in 1920 [19]. Property 7.5 (Clare theorem). [Clare (1908)] If −1 < z < 1 and m, n ∈ N0 , then [19] 

1

−1

P2m (t) P2n+1 (t) dt =

(2n + 1)! (2m)! (−1)m+n . 2(n+m+1)+1 2 (2 (n − m) + 1) (n + m + 1) (n!)2 (m!)2

(7.11)

The result was obtained by Clare in 1908 and indirectly reported by Whittaker and Watson in 1920 [19]. Theorem 7.4 (Catalan theorem). If −1 < z < 1 and n ∈ N0 , then [19] 

1

1 2 (Pn (t) Pn−1 (z) − Pn−1 (t) Pn (z)) dt = − z−t n

(7.12)

   1 d 1 1 = −1. Pn (z) Pn−1 (z) + Pn+1 (z) 2n + 1 dz n n+1

(7.13)

−1

and ∞  n=1

The result was obtained by Catalan and indirectly reported by Whittaker and Watson in 1920 (see [19], p. 332). Theorem 7.5. If −1 < z < 1 and n ∈ N0 , then [229] Pn (z) = Pn(0,0) (z) .

(7.14)

This is the relation between the Legendre and Jacobi polynomials reported by Rainville in 1960 (see [45], p. 273) and Luke in 1969 [14]. Theorem 7.6. [Koepf (1998)] If −1 < z < 1 and n ∈ N0 , then (see [135], p. 22)   1−z . Pn (z) =2 F1 −n, n + 1; 1; 2

(7.15)

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The result was obtained by Koepf in 1998 [135]. As a direct result, we have the following theorem. Theorem 7.7. If −1 < z < 1, α ∈ C, and n ∈ N0 , then    1 − zα Pn zα =2 F1 −n, n + 1; 1; . 2

(7.16)

Theorem 7.8. [Rainville (1960)] If −1 < z < 1 and n ∈ N0 , then [45] Pn (z) =

1 (1,1) (n + 1) Pn−1 (z) . 2

(7.17)

The result was obtained by Rainville in 1960 [45]. Note the following result. Theorem 7.9. If −1 < z < 1, α ∈ C, and n ∈ N0 , then  1 (1,1)  Pn zα = (n + 1) Pn−1 zα . 2

(7.18)

Theorem 7.10. [Rainville (1960) and Freud (1971)] If −1 < t < 1 and n ∈ N0 , then [45,233] ∞ 

− 1  2 Pn (z) t n = 1 − 2zt + t 2 .

(7.19)

n=0

The result is the generating function of the Legendre polynomials obtained by Rainville in 1960 [45] and by Freud in 1971 (see [233], p. 37). At the moment, we easily derive the following result. Theorem 7.11. If −1 < t < 1, α ∈ C, and n ∈ N0 , then − 1   2 Pn zα t n = 1 − 2tzα + t 2

(7.20)

− 1  2 Pn (z) t nα = 1 − 2zt α + t 2α .

(7.21)

∞  n=0

and ∞  n=0

Theorem 7.12 (Hobson theorem). [Hobson (1908)] If α > −1 and n ∈ N, then f (z) =

∞ 

hk Pk (z),

(7.22)

k=0

where

 1  1 hn = k + Pn (z) f (z) dz. 2 −1

(7.23)

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

449

The result is the Hobson series of the Legendre polynomials (also called the Fourier– Legendre series) proposed by Hobson in 1908 [234,235] and further developed by Bojanic and Vuilleumier in 1981 [236] and Rainville in 1960 [45]. Theorem 7.13. [Rainville (1960)] If |z| is sufficiently small and f (z) =

∞ 

sn

n=0

tn , n!

(7.24)

then [45] f (z) =

∞ 

bn Pn (z),

(7.25)

n=0

where sn =

∞  (2n + 1) an+2k   . n+2k k! 3 k=0 2 2

(7.26)

n+k

The result was obtained by Rainville in 1960 [45]. Property 7.6. If −1 < z < 1 and n ∈ N0 , then 

1

−1

(1 + t)−1 (1 − t)−1 Pn (t) dt = 2−1 B (α, β) 3 F2 (−n, n + 1, 0; 1, 0; 1).

(7.27)

The result can be deduced from Rainville [45].

7.1.2 Hypergeometric supertrigonometric functions via Legendre polynomials In the section, we introduce the hypergeometric supertrigonometric functions via Legendre polynomials for the first time. In terms of the Legendre polynomials, we find that   1 + iλz (1)n (7.28) Pn (−iλz) = 2 F1 −n, n + 1; 1; n! 2 and Pn (iλz) =

  1 − iλz (1)n −n, n + 1; 1; , F 2 1 n! 2

(7.29)

where −1 < z < 1, λ ∈ C, and n ∈ N0 . From the expressions of the Legendre polynomials it is not difficult to give the definitions of the new special functions. We now start with the definition of the hypergeometric supersine via Legendre polynomials.

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Definition 7.2. The hypergeometric supersine via Legendre polynomials is defined as      1 + iz 1 − iz (1)n 1 − 2 F1 −n, β + 1; 1; , (7.30) Psinn (z) = 2 F1 −n, n + 1; 1; n! 2i 2 2 where −1 < z < 1 and n ∈ N0 . Definition 7.3. The hypergeometric supercosine via Legendre polynomials is defined as      1 + iz 1 − iz (1)n 1 Pcos n (z) = + 2 F1 −n, n + 1; 1; , (7.31) 2 F1 −n, n + 1; 1; n! 2 2 2 where −1 < z < 1 and n ∈ N0 . Definition 7.4. The hypergeometric supertangent via Legendre polynomials is defined as Ptann (z) =

Psinn (z) , Pcos n (z)

(7.32)

where −1 < z < 1 and n ∈ N0 . Definition 7.5. The hypergeometric supercotangent via Legendre polynomials is defined as Pcos n (z) Pcotann (z) = , (7.33) Psinn (z) where −1 < z < 1 and n ∈ N0 . Definition 7.6. The hypergeometric supersecant via Legendre polynomials is defined as Psec n (z) =

1 , Pcos n (z)

(7.34)

where −1 < z < 1 and n ∈ N0 . Definition 7.7. The hypergeometric supercosecant via Legendre polynomials is defined as Pcosec n (z) =

1 , Psinn (z)

(7.35)

where −1 < z < 1 and n ∈ N0 . In some considerations, we may give the following result. Theorem 7.14. Let −1 < z < 1 and n ∈ N0 . Then we have Pn (iz) = Pcos n (z) + iPsinn (z)

(7.36)

Pn (−iz) = Pcos n (z) − iPsinn (z) .

(7.37)

and

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

451

In view of the expressions of the Legendre polynomials, we can show that Psinn (λz) 1 = (Pn (iλz) − Pn (−iλz)) 2i      1 + iλz 1 − iλz 1 − , −n, n + 1; 1; −n, n + 1; 1; = F F 2 1 2 1 2i 2 2

(7.38)

Pcos n (λz) 1 = (Pn (iλz) + Pn (−iλz)) 2     1 1 + iλz 1 − iλz = + 2 F1 −n, n + 1; 1; , 2 F1 −n, n + 1; 1; 2 2 2

(7.39)

Ptann (λz) Psinn (λz) Pcos n (λz)   1+iλz 2 F1 −n, n + 1; 1; 2  , = 1−iλz 2 F1 −n, n + 1; 1; 2

=

Pcotann (λz) Pcos n (λz) = Psinn (λz)   1−iλz 2 F1 −n, n + 1; 1; 2  , = 1+iλz 2 F1 −n, n + 1; 1; 2 Psec n (λz) =

1 , Pcos n (λz)

(7.40)

(7.41)

(7.42)

and Pcosec n (λz) = where i =

√ −1, −1 < z < 1, λ ∈ C, and n ∈ N0 .

1 , Psinn (λz)

(7.43)

Definition 7.8. The hypergeometric supersine Psinn (z), hypergeometric supercosine Pcos n (z), hypergeometric supertangent Ptann (z), hypergeometric supercotangent Pcotann (z), hypergeometric supersecant Psec n (z), and hypergeometric supercosecant Pcosec n (z) are called the hypergeometric supertrigonometric functions via Legendre polynomials. Furthermore, we present the following theorems in terms of the hypergeometric supersine and the hypergeometric supercosine.

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Theorem 7.15. If i = as follows:

√ −1, t ∈ C, −1 < z < 1, and k ∈ N0 , then the generating functions are ∞ 

Psink (z) t k =

k=0

 1  −1 × ς1 − ς2−1 2i

(7.44)

and ∞ 

  Pcos k (z) t k = 2−1 × ς1−1 + ς2−1 ,

(7.45)

k=0

where 1  2 ς1 = 1 − 2izt + t 2

(7.46)

1  2 ς2 = 1 + 2izt + t 2 .

(7.47)

and

Proof. By means of the series representations ∞ 

Pk (iz) t k = ς1−1

(7.48)

Pk (−iz) t k = ς2−1 ,

(7.49)

1  2 ς1 = 1 − 2izt + t 2

(7.50)

1  2 ς2 = 1 + 2izt + t 2 ,

(7.51)

k=0

and ∞  k=0

where

and

we obtain ∞ 

Psink (z) t k =

k=0

∞  1  1  −1 × ς1 − ς2−1 (Pk (iz) − Pk (−iz)) t k = 2i 2i

(7.52)

  1 (Pk (iz) + Pk (−iz)) t k = 2−1 × ς1−1 + ς2−1 . 2

(7.53)

k=0

and ∞  k=0

∞

Pcos k (z) t k =

k=0

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

453

7.1.3 Hypergeometric superhyperbolic functions via Legendre polynomials In this section, we present the hypergeometric superhyperbolic functions via Legendre polynomials. Now let us consider the particular cases of the Legendre polynomials with parameters as follows:   1 + λz Pn (−λz) = 2 F1 −n, n + 1; 1; (7.54) 2 and

  1 − λz , Pn (λz) = 2 F1 −n, n + 1; 1; 2

(7.55)

where −1 < z < 1, λ ∈ C, and n ∈ N0 . By employing the above relations we define the special functions containing the Legendre polynomials as follows. Definition 7.9. The hypergeometric superhyperbolic supersine via Legendre polynomials is defined as      1+z 1−z 1 Psinhn (z) = −n, n + 1; 1; −n, n + 1; 1; − , (7.56) F F 2 1 2 1 2 2 2 where −1 < z < 1 and n ∈ N0 . Definition 7.10. The hypergeometric superhyperbolic supercosine via Legendre polynomials is defined as      1+z 1−z 1 + 2 F1 −n, n + 1; 1; , (7.57) Pcoshn (z) = 2 F1 −n, n + 1; 1; 2 2 2 where −1 < z < 1 and n ∈ N0 . Definition 7.11. The hypergeometric superhyperbolic supertangent via Legendre polynomials is defined as Ptanhn (z) =

Psinhn (z) , Pcoshn (z)

(7.58)

where −1 < z < 1 and n ∈ N0 . Definition 7.12. The hypergeometric superhyperbolic supercotangent via Legendre polynomials is defined as Pcotanhn (z) = where −1 < z < 1 and n ∈ N0 .

Pcoshn (z) , Psinhn (z)

(7.59)

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Definition 7.13. The hypergeometric superhyperbolic supersecant via Legendre polynomials is defined as Psechn (z) =

1 , Pcoshn (z)

(7.60)

where −1 < z < 1 and n ∈ N0 . Definition 7.14. The hypergeometric superhyperbolic supercosecant via Legendre polynomials is defined as Pcosechn (z) =

1 , Psinhn (z)

(7.61)

where −1 < z < 1 and n ∈ N0 . Without proof, we directly have the following theorem. Theorem 7.16. We have Pn (z) = Pcoshn (z) + Psinhn (z)

(7.62)

Pn (−z) = Pcoshn (z) − Psinhn (z) ,

(7.63)

and

where −1 < z < 1 and n ∈ N0 . By evaluating the new special functions, we have that Psinhn (λz) 1 = (Pn (λz) − Pn (−λz)) 2     1 + λz 1 − λz 1 −n, n + 1; 1; −n, n + 1; 1; = − , F F 2 1 2 1 2 2 2

(7.64)

Pcoshn (λz) 1 = (Pn (λz) + Pn (−λz)) 2     1 1 + λz 1 − λz = + 2 F1 −n, n + 1; 1; , 2 F1 −n, n + 1; 1; 2 2 2

(7.65)

Ptanhn (λz) Psinhn (λz) Pcoshn (λz)   1+λz 2 F1 −n, n + 1; 1; 2  , = 1−λz 2 F1 −n, n + 1; 1; 2 =

(7.66)

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

455

Pcotanhn (λz) Pcoshn (λz) Psinhn (λz)   1−λz 2 F1 −n, n + 1; 1; 2  , = 1+λz 2 F1 −n, n + 1; 1; 2

=

Psechn (λz) =

1 , P coshn (λz)

(7.67)

(7.68)

and Pcosechn (λz) = where i =

√ −1, −1 < z < 1, λ ∈ C, and n ∈ N0 .

1 , P sinhn (λz)

(7.69)

Definition 7.15. The hypergeometric superhyperbolic supersine Psinhn (z), hypergeometric superhyperbolic supercosine Pcoshn (z), hypergeometric superhyperbolic supertangent Ptanhn (z), hypergeometric superhyperbolic supercotangent Pcotanhn (z), hypergeometric superhyperbolic supersecant Psechn (z), and hypergeometric superhyperbolic supercosecant Pcosechn (z) are called the hypergeometric superhyperbolic functions via Legendre polynomials. Theorem 7.17. If t ∈ C, −1 < z < 1, and k ∈ N0 , then we have the following generating functions: ∞    (7.70) Psinhk (z) t k = 2−1 ς3−1 − ς4−1 k=0

and ∞ 

  Pcoshk (z) t k = 2−1 ς3−1 + ς4−1 ,

(7.71)

k=0

where 1  2 ς3 = 1 − 2zt + t 2

(7.72)

1  2 ς4 = 1 + 2zt + t 2 .

(7.73)

and

Proof. With the series representations ∞  k=0

Pk (z) t k = ς3−1

(7.74)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and ∞ 

Pk (−z) t k = ς4−1 ,

(7.75)

1  2 ς3 = 1 − 2zt + t 2

(7.76)

1  2 ς4 = 1 + 2zt + t 2 ,

(7.77)

k=0

where

and

we have that ∞ 

  1 (Pk (z) − Pk (−z)) t k = 2−1 ς3−1 − ς4−1 2

(7.78)

  1 (Pk (z) + Pk (−z)) t k = 2−1 × ς3−1 + ς4−1 . 2

(7.79)

∞

Psinhk (z) t k =

k=0

k=0

and ∞ 

∞

Pcoshk (z) t k =

k=0

k=0

7.2 Legendre-type polynomials In the section, we present the Legendre type polynomials for the first time.

7.2.1 Definition, properties, and theorems for the Legendre-type polynomials We now start with the definition for the Legendre type polynomials. Definition 7.16. The Legendre-type polynomial of degree n is defined by n (z) = 2 F1 (−n, n + 1; 1; z) = (−1)

n

(7.80)

2 F1 (−n, n + 1; 1; z + 1),

where 0 < z < 1 and n ∈ N0 . Evidently, we have that

 Pn (z) = n

where 0 < z < 1 and n ∈ N0 .

 1−z , 2

(7.81)

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

457

Definition 7.17. The Legendre-type polynomial of degree n with power law is defined as  n z α  (7.82) = 2 F1 −n, n + 1; 1; zα  n α = (−1) 2 F1 −n, n + 1; 1; z + 1 , where 0 < α, z < 1 and n ∈ N0 . For z = 1, we have that n (1) =

(−n)n n!

(7.83)

since (−n)n (1)n , 2 F1 (−n, n + 1; 1; 1) = n! n!

(7.84)

where 2 F1 (−n, n + 1; 1; 1) =

(−n)n . (1)n

(7.85)

Similarly, we have the following results. Theorem 7.18. Let 0 < z < 1 and n ∈ N0 . Then  1 1 n (z) = t n (1 − t)−(n+1) 1 F0 (−n; −; zt)dt. (n!)2 0

(7.86)

Proof. By means of the relation (see Chapter 2) 2 F1 (−n, n + 1; 1; z)

 1 1 = t n (1 − t)−(n+1) 1 F0 (−n; −; zt)dt  (n + 1)  (1 + n) 0  1 1 t n (1 − t)−(n+1) 1 F0 (−n; −; zt)dt = (n!)2 0

(7.87)

we obtain that n (z) (1)n = 2 F1 (−n, n + 1; 1; z) n!  (1)n 1 n t (1 − t)−(n+1) 1 F0 (−n; −; zt)dt = (n!)3 0  1 1 t n (1 − t)−(n+1) 1 F0 (−n; −; zt)dt. = (n!)2 0

(7.88)

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Without proof, we present the following theorem. Theorem 7.19. If n ∈ N0 , Re (c) > Re (μ) > 0, and |z| < 1, then  2 F1 (−n, n + 1; 1 + μ; z) = μ

1

(1 − t)μ−1 n (zt)dt.

(7.89)

0

Theorem 7.20. Let 0 < z < 1 and n ∈ N0 . Then the Legendre-type polynomial is the solution of the differential equation z (1 − z)

d 2ϕ dϕ + n (n + 1) ϕ = 0. + (1 − 2z) dz dz2

(7.90)

Theorem 7.21. Let λ, z ∈ C, 0 < z < 1, and n ∈ N0 . Then we have  

  (λ) λ, −n, n + 1 z λ−1 ; , n (zt) = λ 3 F1 L t 1 s s

(7.91)

where the Laplace transform of a function f is  L [f (t)] = f (s) =

∞

e−st f (t) dt.

(7.92)

0

Proof. By means of the formula (Chapter 3) we directly obtain that

 L t λ−1 n (zt) 

= L t λ−1 2 F1 (−n, n + 1; 1; zt)     ∞ a, b = e−st t λ−1 2 F1 ; zt dt c 0    (λ) λ, −n, n + 1 z . = λ 3 F1 ; 1 s s

(7.93)

Theorem 7.22. If −1 < z < 1 and k ∈ N0 , then the generating function is ∞ 

− 1  2 k (z) t k = 1 − 2 (1 − 2z) t + t 2 .

(7.94)

k=0

Proof. By means of the above result the equality can be deduced from the Legendre-type polynomials. We now present the Legendre-type polynomial.

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

459

Definition 7.18. The Legendre-type polynomial n (z) can be written as n (z) =

n  k=0

 2 (1)n t k (1 − t)n−k , 2n k! (n − k)! (1)k (1)n−k

(7.95)

where −1 < z < 1 and n ∈ N0 . By this definition we derive the following theorems. Theorem 7.23. Let −1 < z < 1 and n ∈ N0 . Then we have the Rodrigues formula for the Legendre-type polynomial n (z): n (z) =

1 dn  n z (1 − z)n . n n! dz

(7.96)

Theorem 7.24. If −1 < λz < 1, λ ∈ R, and n ∈ N0 , then 

z

t c−1 (z − t)d−1 n (λt) dt = zc+d−1 B (c, d) 3 F2 (−n, n + 1, c; 1, c + d; λz).

(7.97)

0

Theorem 7.25. If c, d ∈ C\Z and n ∈ N0 , then 

1

t c−1 (1 − t)d−1 n (t) dt = B (c, d) 3 F2 (−n, n + 1, c; 1, c + d; 1).

(7.98)

0

Theorem 7.26. If −1 < λz < 1, c, d ∈ C\Z, λ ∈ R, and n ∈ N0 , then 

z

t c−1 (z − t)d−1 n (λt) dt = zc+d−1 B (c, d) 3 F2 (−n, n + 1, c; 1, c + d; λz).

(7.99)

0

Theorem 7.27. If c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then 

1

t c−1 (1 − t)d−1 n (λt) dt = B (c, d) 3 F2 (−n, n + 1, c; 1, c + d; λ).

(7.100)

0

Theorem 7.28. If c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then 

1

t c−1 (1 − t)d−1 n (λt) dt = B (c, d) 3 F2 (−n, n + 1, c; 1, c + d; λ).

(7.101)

0

Theorem 7.29. If c, d ∈ C\Z, −1 < λz < 1, λ ∈ R, and n ∈ N0 , then 

1 −1

(1 + t)c−1 (1 + t)d−1 n (λ (1 + t)) dt = 2c+d−1

× B (c, d) 3 F2 (−n, n + 1, c; 1, c + d; 2λ).

(7.102)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 7.30. If c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then  z   t c−1 (z − t)d−1 n λt 2 dt = zc+d−1 B (c, d) 0   c c+1 c+d c+d +1 2 ; 1, , ; λz . × 4 F3 −n, n + 1, , 2 2 2 2

(7.103)

Theorem 7.31. If c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then 

  t c−1 (1 − t)d−1 n λt 2 dt = B (c, d) 0   c c+1 c+d c+d +1 ; 1, , ;λ . × 4 F3 −n, n + 1, , 2 2 2 2 1

(7.104)

Theorem 7.32. If c, d ∈ C\Z, −1 < λ < 1, λ ∈ R, and n ∈ N0 , then 

  (1 + t)c−1 (1 − t)d−1 n λ (1 + t)2 dt = 2c+d−1 B (c, d) −1   c c+1 c+d c+d +1 ; 1, , ; 4λ . × 4 F3 −n, n + 1, , 2 2 2 2 1

Theorem 7.33. If c, d ∈ C\Z, −1 < z < 1, and n ∈ N0 , then  z t c−1 (z − t)d−1 n (λ (z − t)) dt = zc+d−1 B (c, d) 0

(7.105)

(7.106)

× 3 F2 (−n, n + 1, d; 1, c + d; λz). Theorem 7.34. If −1 < z < 1 and n ∈ N0 , then 

1

t c−1 (1 − t)d−1 n (λ (1 − t)) dt = B (c, d)

0

(7.107)

× 3 F2 (−n, n + 1, d; 1, c + d; λ). Theorem 7.35. If −1 < z < 1 and n ∈ N0 , then 

1 −1

(1 + t)c−1 (1 − t)d−1 n (λ (1 − t)) dt = 2c+d−1 B (c, d)

(7.108)

× 3 F2 (−n, n + 1, d; 1, c + d; 2λ). Theorem 7.36. If −1 < z < 1 and n ∈ N0 , then  z t c−1 (z − t)d−1 n (λt (z − t)) dt = zc+d−1 B (c, d) 0   c + d c + d + 1 λz2 × 4 F3 −n, n + 1, c, d; 1, , ; . 2 2 4

(7.109)

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

461

Theorem 7.37. If −1 < z < 1 and n ∈ N0 , then 

1

t c−1 (1 − t)d−1 n (λt (1 − t)) dt = B (c, d)   c+d c+d +1 λ , ; . × 4 F3 −n, n + 1, c, d; 1, 2 2 4 0

(7.110)

Theorem 7.38. If −1 < z < 1 and n ∈ N0 , then 

1

(1 + t)c−1 (1 − t)d−1 n (λ (1 + t) (1 − t)) dt = 2c+d−1 B (c, d)   c+d c+d +1 , ;λ . × 4 F3 −n, n + 1, c, d; 1, 2 2 −1

(7.111)

7.2.2 Hypergeometric supertrigonometric via Legendre-type polynomials In the section, we report the hypergeometric supertrigonometric via Legendre-type polynomials. Now from the definition of the Legendre-type polynomials we easily verify that n (−iλz) = 2 F1 (−n, n + 1; 1; −iλz)

(7.112)

n (iλz) = 2 F1 (−n, n + 1; 1; iλz),

(7.113)

and

where −1 < z < 1, λ ∈ C, and n ∈ N0 . Definition 7.19. The hypergeometric supersine via Legendre-type polynomials is defined as sinn (z) = 2 Supersin1 (−n, n + 1; 1; z),

(7.114)

where −1 < z < 1 and n ∈ N0 . Definition 7.20. The hypergeometric supercosine via Legendre-type polynomials is defined as cos n (z) = 2 Supercos 1 (−n, n + 1; 1; z),

(7.115)

where −1 < z < 1 and n ∈ N0 . Definition 7.21. The hypergeometric supertangent via Legendre-type polynomials is defined as sinn (z) tann (z) = , (7.116) cos n (z) where −1 < z < 1 and n ∈ N0 .

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Definition 7.22. The hypergeometric supercotangent via Legendre-type polynomials is defined as cos n (z) , (7.117) cotann (z) = sinn (z) where −1 < z < 1 and n ∈ N0 . Definition 7.23. The hypergeometric supersecant via Legendre-type polynomials is defined as 1 sec n (z) = , (7.118) cos n (z) where −1 < z < 1 and n ∈ N0 . Definition 7.24. The hypergeometric supercosecant via Legendre-type polynomials is defined as 1 cosec n (z) = , (7.119) sinn (z) where −1 < z < 1 and n ∈ N0 . Based on the above results, we obtain the following theorem. Theorem 7.39. We have n (iz) = cos n (z) + isinn (z)

(7.120)

n (−iz) = cos n (z) − isinn (z) ,

(7.121)

and

where −1 < z < 1 and n ∈ N0 . By the definitions of the new special functions we have that sinn (λz) 1 = (n (iλz) − n (−iλz)) 2i 1 = (2 F1 (−n, n + 1; 1; iλz) − 2 F1 (−n, n + 1; 1; iλz)) 2i = 2 Supersin1 (−n, n + 1; 1; λz) ,

(7.122)

cos n (λz) 1 = (n (iλz) + n (−iλz)) 2 1 = (2 F1 (−n, n + 1; 1; iλz) − 2 F1 (−n, n + 1; 1; −iλz)) 2 = 2 Supercos 1 (−n, n + 1; 1; λz) ,

(7.123)

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

463

tann (λz) sinn (λz) cos n (λz) 2 Supersin1 (−n, n + 1; 1; λz) = 2 Supercos 1 (−n, n + 1; 1; λz)

=

(7.124)

= 2 Supertan1 (−n, n + 1; 1; λz), cotann (λz) cos n (λz) = sinn (λz) 2 Supercos 1 (−n, n + 1; 1; λz) = 2 Supersin1 (−n, n + 1; 1; λz)

(7.125)

= 2 Supercotan1 (−n, n + 1; 1; λz), sec n (λz) =

1 , cos n (λz)

(7.126)

and cosec n (λz) =

1 , sinn (λz)

(7.127)

√ where i = −1, −1 < z < 1, λ ∈ C, and n ∈ N0 . So, we easily give the definition of the hypergeometric supertrigonometric functions via Legendre-type polynomials. Definition 7.25. The hypergeometric supersine sinn (z), hypergeometric supercosine cos n (z), hypergeometric supertangent tann (z), hypergeometric supercotangent cotann (z), hypergeometric supersecant sec n (z), and hypergeometric supercosecant cosec n (z) are called the hypergeometric supertrigonometric functions via Legendre-type polynomials. Proceeding again in a similar manner, we obtain the following theorems. Theorem 7.40. If −1 < z < 1, −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 , then 

z

t c−1 (z − t)d−1 sinn (λt) dt = zc+d−1 B (c, d)

0

(7.128)

× 3 Supersin2 (−n, n + 1, c; 1, c + d; λz). Proof. With the representations sinn (z) =

1 (n (iλz) − n (−iλz)) 2i

(7.129)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and (see Chapter 2) 

z

t c−1 (z − t)d−1 2 F1 (−n, n + 1; 1; λt)dt

0

(7.130)

= zc+d−1 B (c, d) 3 F2 (−n, n + 1, c; 1, c + d; λz), we have



z

t c−1 (z − t)d−1 sinn (λt) dt 0  z t c−1 (z − t)d−1 2 Supersin1 (−n, n + 1; 1; λt)dt =

(7.131)

0

= zc+d−1 B (c, d) 3 Supersin2 (−n, n + 1, c; 1, c + d; λz), where −1 < z < 1, −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 . Without proofs, we easily derive the following theorems. Theorem 7.41. If −1 < z < 1, −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 , then  z t c−1 (z − t)d−1 sinn (t) dt 0

(7.132)

= zc+d−1 B (c, d) 3 Supersin2 (−n, n + 1, c; 1, c + d; z). Theorem 7.42. If −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 , then  1 t c−1 (1 − t)d−1 sinn (λt) dt = B (c, d) 0

(7.133)

× 3 Supersin2 (−n, n + 1, c; 1, c + d; λ). Based on the above results, we derive the following theorem. Theorem 7.43. If −1 < z < 1, −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 , then  z t c−1 (z − t)d−1 cos n (λt) dt = zc+d−1 B (c, d) 0

(7.134)

× 3 Supercos 2 (−n, n + 1, c; 1, c + d; λz). Proof. Putting the formulas cos n (z) =

1 (n (iλz) + n (−iλz)) 2

(7.135)

and (see Chapter 2) 

z

t c−1 (z − t)d−1 2 F1 (−n, n + 1; 1; λt)dt

0

=z

c+d−1

B (c, d) 3 F2 (−n, n + 1, c; 1, c + d; λz),

(7.136)

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

we have



465

z

t c−1 (z − t)d−1 cos n (λt) dt  z t c−1 (z − t)d−1 2 Supercos 1 (−n, n + 1; 1; λt)dt = 0

(7.137)

0

= zc+d−1 B (c, d) 3 Supercos 2 (−n, n + 1, c; 1, c + d; λz), where −1 < z < 1, −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 . Proceeding in the same way, we obtain the following theorems without proofs. Theorem 7.44. If −1 < z < 1, c, d ∈ C\Z− , and n ∈ N0 , then  z t c−1 (z − t)d−1 cos n (t) dt = zc+d−1 B (c, d) 3 Supercos 2 (−n, n + 1, c; 1, c + d; z). (7.138) 0

Theorem 7.45. If c, d ∈ C\Z− and n ∈ N0 , then  1 t c−1 (1 − t)d−1 cos n (t) dt = B (c, d) 3 Supercos 2 (−n, n + 1, c; 1, c + d; 1).

(7.139)

0

Theorem 7.46. If −1 < z < 1, −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 , then  1 t c−1 (1 − t)d−1 cos n (λt) dt = B (c, d) 3 Supercos 2 (−n, n + 1, c; 1, c + d; λ).

(7.140)

0

7.2.3 Hypergeometric superhyperbolic functions via Legendre-type polynomials In the section, we consider the hypergeometric superhyperbolic functions via Legendretype polynomials. In a similar way, let us write the formulas n (λz) = 2 F1 (−n, n + 1; 1; λz)

(7.141)

n (−λz) = 2 F1 (−n, n + 1; 1; −λz),

(7.142)

and

where −1 < z < 1 and n ∈ N0 . From the above formulations we introduce the following definitions for the special functions. Definition 7.26. The hypergeometric superhyperbolic supersine via Legendre-type polynomials is defined as sinhn (z) = 2 Supersinh1 (−n, n + 1; 1; z), where −1 < z < 1 and n ∈ N0 .

(7.143)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Definition 7.27. The hypergeometric superhyperbolic supercosine via Legendre-type polynomials is defined as coshn (z) = 2 Supercosh1 (−n, n + 1; 1; z),

(7.144)

where −1 < z < 1 and n ∈ N0 . Definition 7.28. The hypergeometric superhyperbolic supertangent via Legendre type polynomials is defined as tanhn (z) =

sinhn (z) , coshn (z)

(7.145)

where −1 < z < 1 and n ∈ N0 . Definition 7.29. The hypergeometric superhyperbolic supercotangent via Legendre-type polynomials is defined as cotanhn (z) =

coshn (z) , sinhn (z)

(7.146)

where −1 < z < 1 and n ∈ N0 . Definition 7.30. The hypergeometric superhyperbolic supersecant via Legendre-type polynomials is defined as sechn (z) =

1 , coshn (z)

(7.147)

where −1 < z < 1 and n ∈ N0 . Definition 7.31. The hypergeometric superhyperbolic supercosecant via Legendre-type polynomials is defined as cosechn (z) =

1 , sinhn (z)

(7.148)

where −1 < z < 1 and n ∈ N0 . From the above formulae we get the series representations sinhn (z) 1 = (n (λz) − n (−λz)) 2 1 = (2 F1 (−n, n + 1; 1; λz) − 2 F1 (−n, n + 1; 1; λz)) 2 = 2 Supersinh1 (−n, n + 1; 1; λz) ,

(7.149)

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

coshn (λz) 1 = (Rn (λz) + Rn (−λz)) 2 1 = (2 F1 (−n, n + 1; 1; λz) + 2 F1 (−n, n + 1; 1; −λz)) 2 = 2 Supercosh1 (−n, n + 1; 1; λz) ,

467

(7.150)

tanhn (λz) sinhn (λz) coshn (λz) 2 Supersinh1 (−n, n + 1; 1; λz) = 2 Supercosh1 (−n, n + 1; 1; λz)

=

(7.151)

= 2 Supertanh1 (−n, n + 1; 1; λz), cotanhn (λz) coshn (λz) sinhn (λz) 2 Supercosh1 (−n, n + 1; 1; λz) = 2 Supersinh1 (−n, n + 1; 1; λz)

=

(7.152)

= 2 Supercotanh1 (−n, n + 1; 1; λz), sechn (λz) =

1 , coshn (λz)

(7.153)

and cosechn (λz) =

1 , sinhn (λz)

(7.154)

√ where i = −1, −1 < z < 1, −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 . With the results, we easily present the following definition. Definition 7.32. The hypergeometric superhyperbolic supersine sinhn (z), hypergeometric superhyperbolic supercosine coshn (z), hypergeometric superhyperbolic supertangent tanhn (z), hypergeometric superhyperbolic supercotangent cotanhn (z), hypergeometric superhyperbolic supersecant sechn (z), and hypergeometric superhyperbolic supercosecant cosechn (z) are called the hypergeometric superhyperbolic functions via Legendre-type polynomials. In this part, applying the above formulae, we easily derive the following theorems without proofs. Theorem 7.47. If −1 < z < 1, −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 , then  z t c−1 (z − t)d−1 sinhn (λt) dt = zc+d−1 B (c, d) 0

× 3 Supersinh2 (−n, n + 1, c; 1, c + d; λz).

(7.155)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

Theorem 7.48. If −1 < z < 1, c, d ∈ C\Z− , and n ∈ N0 , then  z t c−1 (z − t)d−1 sinhn (t) dt 0

(7.156)

= zc+d−1 B (c, d) 3 Supersinh2 (−n, n + 1, c; 1, c + d; z). Theorem 7.49. If c, d ∈ C\Z− and n ∈ N0 , then  1 t c−1 (1 − t)d−1 sinhn (t) dt = B (c, d) 0

(7.157)

× 3 Supersinh2 (−n, n + 1, c; 1, c + d; 1). Theorem 7.50. If −1 < z < 1, −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 , then  1 t c−1 (1 − t)d−1 sinhn (λt) dt = B (c, d) 0

(7.158)

× 3 Supersinh2 (−n, n + 1, c; 1, c + d; λ). Theorem 7.51. If −1 < z < 1, −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 , then  z t c−1 (z − t)d−1 coshn (λt) dt = zc+d−1 B (c, d) 0

(7.159)

× 3 Supercosh2 (−n, n + 1, c; 1, c + d; λz). Theorem 7.52. If −1 < z < 1, −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 , then  z t c−1 (z − t)d−1 coshn (t) dt 0

(7.160)

= zc+d−1 B (c, d) 3 Supercosh2 (−n, n + 1, c; 1, c + d; z). Theorem 7.53. If −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 , then  1 t c−1 (1 − t)d−1 coshn (t) dt = B (c, d) 0

(7.161)

× 3 Supercosh2 (−n, n + 1, c; 1, c + d; 1). Theorem 7.54. If −1 < λ < 1, c, d ∈ C\Z− , and n ∈ N0 , then  1 t c−1 (1 − t)d−1 coshn (λt) dt = B (c, d) 0

(7.162)

× 3 Supercosh2 (−n, n + 1, c; 1, c + d; λ). In some considerations, we get n (z) = coshn (z) + sinhn (z) , where −1 < z < 1.

(7.163)

Chapter 7 • Hypergeometric supertrigonometric and superhyperbolic functions

469

By means of the above definitions, we conclude the following theorem without proof. Theorem 7.55. Let −1 < z < 1 and n ∈ N0 . Then we have sinn (z) = Psinn (−i − 2z) ,

(7.164)

cos n (z) = Pcos n (−i − 2z) ,

(7.165)

tann (z) = Ptann (−i − 2z) ,

(7.166)

cotann (z) = Pcotann (−i − 2z) ,

(7.167)

sec n (z) = Psec n (−i − 2z) ,

(7.168)

cosec n (z) = Pcosec n (−i − 2z) ,

(7.169)

sinhn (z) = Psinhn (−i − 2z) ,

(7.170)

coshn (z) = Pcoshn (−i − 2z) ,

(7.171)

tanhn (z) = Pcoshn (−i − 2z) ,

(7.172)

cotanhn (z) = Pcoshn (−i − 2z) ,

(7.173)

sechn (z) = Psechn (−i − 2z) ,

(7.174)

cosechn (z) = Pcosechn (−i − 2z) ,   z+i = Psinn (z) , sinn − 2   z+i = Pcos n (z) , cos n − 2   z+i = Ptann (z) , tann − 2   z+i = Pcotann (z) , cotann − 2   z+i = Psec n (z) , sec n − 2   z+i = Pcosec n (z) , cosec n − 2   z+i = Psinhn (z) , sinhn − 2   z+i = Pcoshn (z) , coshn − 2   z+i = Pcoshn (z) , tanhn − 2   z+i = Pcoshn (z) , cotanhn − 2

(7.175) (7.176) (7.177) (7.178) (7.179) (7.180) (7.181) (7.182) (7.183) (7.184) (7.185)

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An Introduction to Hypergeometric, Supertrigonometric, Superhyperbolic Functions

and

  z+i sechn − = Psechn (z) , 2

(7.186)

  z+i = Pcosechn (z) . cosechn − 2

(7.187)

Furthermore, we can establish the following series representations. Theorem 7.56. If −1 < t < 1, −1 < z < 1, and k ∈ N0 , then we have the following generating functions: ∞ 

sink (z) t k =

k=0

 1  −1 ϑ1 − ϑ2−1 2i

(7.188)

and ∞ 

  cos k (z) t k = 2−1 × ϑ1−1 + ϑ2−1 ,

(7.189)

k=0

where

and

1  2 ϑ1 = 1 − 2 (1 − 2iz) t + t 2

(7.190)

1  2 ϑ2 = 1 − 2 (1 + 2iz) t + t 2 ,

(7.191)

Theorem 7.57. If −1 < t < 1, −1 < z < 1, and k ∈ N0 , then we have the following generating functions: sinhk (z) t k =

 1  −1 ϑ3 − ϑ4−1 2

(7.192)

coshk (z) t k =

 1  −1 ϑ3 + ϑ4−1 , 2

(7.193)

∞  k=0

and ∞  k=0

where

and

1  2 ϑ3 = 1 − 2 (1 − 2z) t + t 2

(7.194)

1  2 ϑ4 = 1 − 2 (1 + 2z) t + t 2 .

(7.195)

For more results on the Legendre polynomials, see Abramowitz and Stegun (1948) [125], Erdelyi et al. (1953) [88], Hobson (1955) [237], Robin (1957) [238], and Snow (1952) [239].

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Index B Barnes contour integral, 144 Bateman theorem, 142, 296 Beta function Euler, 9, 12 incomplete, 9 incomplete, 11 Brafman path, 347 theorem, 347, 392 C Carlitz theorem, 298 Catalan theorem, 447 Chu–Vandermonde identity, 140 Clare theorem, 447 Clausen hypergeometric functions, 13, 115 series, 13–18, 47, 49–55, 57, 73–75, 81, 97–100, 104–109, 136, 137 Confluent hypergeometric differential equation, 238 functions, 266 Contour Hankel, 11 integral Barnes, 144 loop, 11 Convergence, 14, 49–52, 57, 74, 75, 81, 82, 97–100, 105–108, 139 Cosecant hypergeometric superhyperbolic, 75, 76, 108, 151, 162, 164, 255, 256, 279 Cosine hypergeometric, 148 superhyperbolic, 74, 76, 106, 108, 137, 150–153, 162, 164, 165, 169, 254, 256, 257, 279

Cotangent hypergeometric superhyperbolic, 75, 76, 107, 108, 150, 151, 162, 164, 255, 256, 279 D Denominator parameters, 97–102, 104–109 Differential equation confluent hypergeometric, 238 for the Clausen hypergeometric series, 14 Gauss, 142, 143, 180, 181, 185 Laguerre, 346 linear homogeneous, 297 Dilogarithm, 112–114 E Euler completion formula, 4 constant, 1 function, 6, 12 beta, 9, 12 gamma, 1, 12 functional equation, 2 integral of first kind, 1 of second kind, 10 theorem, 2, 139 limit, 7 F Feldheim theorem, 349 Formula Euler completion, 4 Gauss multiplication, 4 Legendre duplication, 4 Rodrigues, 294, 445, 459 Schläfli’s integral, 446 Fourier–Gauss-type hypergeometric series, 147

481

482

Index

Fourier–Jacobi series, 309 Fourier–Legendre series, 449 Function Euler, 6, 12 beta, 9, 12 gamma, 1, 12 gamma, 1, 11 hypergeometric, 53 Clausen, 13, 115 confluent, 266 Gauss, 203 superhyperbolic, 73, 74, 76, 81, 105, 108, 109, 149, 151, 153, 169, 254, 256, 257, 276, 277, 279, 280, 304, 307, 330, 332, 367, 368, 401, 402, 407, 408, 413, 415, 453, 455, 465, 467 supertrigonometric, 49, 50, 53–55, 57, 97, 101, 102, 104, 105, 145, 146, 148, 242, 243, 245, 246, 267–272, 300, 303, 320, 323, 356, 357, 359, 361, 398, 399, 403, 405, 410, 412, 449, 451, 463 incomplete beta, 11 Laguerre, 372 Luke generating, 310 Rainville, 391, 398, 399, 401, 402 Szegö of first type, 390, 403–405, 407, 408 of second type, 391, 410–415 G Gamma function, 1, 11 Euler, 1, 12 incomplete, 1 incomplete, 5, 6, 284 Gauss differential equation, 142, 143, 180, 181, 185 hypergeometric functions, 203 series, 13, 139, 142, 144–154, 169, 181, 233, 293 multiplication formula, 4 theorem, 4 theorem, 140 Generalized hypergeometric series, 13

Glaisher theorem, 446 H Hankel contour, 11 integral theorem, 11 Hille theorem, 347 Hobson series, 449 theorem, 448 Hypergeometric, 13 cosine, 148 functions, 53 Clausen, 13, 115 confluent, 266 Gauss, 203 series, 13, 14, 54, 79, 111, 142, 143, 180, 181, 185, 238, 256 Clausen, 13–18, 47, 49–55, 57, 73–75, 81, 97–100, 104–109, 136, 137 Fourier–Gauss-type, 147 Gauss, 13, 139, 142, 144–154, 169, 181, 233, 293 generalized, 13 Kummer confluent, 237, 242–246, 254–257, 265, 267–272, 276–280, 284, 291 sine, 147, 148, 243 supercosecant, 52, 53, 101, 146, 161, 163, 243–245, 269, 270, 302, 303, 321, 323, 357, 399, 405, 411, 412, 415, 450, 451, 462, 463 supercosine, 51–53, 98, 101, 136, 145, 146, 149, 161, 163, 164, 169, 243–245, 267, 268, 270, 301, 303, 321, 323, 356, 357, 398, 399, 404, 405, 410, 412, 415, 450, 451, 461, 463 supercotangent, 52, 53, 99, 101, 145, 146, 161, 163, 243–245, 268, 270, 301, 303, 321, 323, 357, 398, 399, 404, 405, 411, 412, 415, 450, 451, 462, 463 superhyperbolic cosecant, 75, 76, 108, 151, 162, 164, 255, 256, 279

Index

cosine, 74, 76, 106, 108, 137, 150–153, 162, 164, 165, 169, 254, 256, 257, 279 cotangent, 75, 76, 107, 108, 150, 151, 162, 164, 255, 256, 279 functions, 73, 74, 76, 81, 105, 108, 109, 149, 151, 153, 169, 254, 256, 257, 276, 277, 279, 280, 304, 307, 330, 332, 367, 368, 401, 402, 407, 408, 413, 415, 453, 455, 465, 467 secant, 75, 76, 107, 108, 137, 150, 151, 153, 162, 164, 255, 256, 279 sine, 74, 76, 105, 108, 137, 149–153, 161, 163, 165, 169, 254, 256, 257, 279 supercosecant, 277, 279, 306, 307, 331, 332, 368, 402, 408, 415, 454, 455, 466, 467 supercosine, 276, 278, 305, 307, 330, 332, 367, 368, 401, 402, 407, 408, 414, 453, 455, 466, 467 supercotangent, 277, 278, 305, 307, 331, 332, 368, 401, 402, 408, 414, 453, 455, 466, 467 supersecant, 277, 279, 306, 307, 331, 332, 368, 402, 408, 415, 454, 455, 466, 467 supersine, 276, 278, 305, 307, 330, 332, 367, 368, 385, 401, 402, 407, 408, 414, 453, 455, 465, 467 supertangent, 276, 278, 305, 307, 330, 332, 368, 401, 402, 407, 408, 414, 453, 455, 466, 467 tangent, 74, 76, 107, 108, 137, 150–152, 162, 164, 255, 256, 279 supersecant, 52, 53, 99–101, 137, 146, 148, 161, 163, 243–245, 268, 270, 301, 303, 321, 323, 357, 399, 405, 411, 412, 415, 450, 451, 462, 463 supersine, 50, 53, 97, 101, 136, 145, 146, 148, 149, 160, 162, 164, 169, 243–245, 267–270, 301, 303, 320, 323, 356, 357, 362, 398, 399, 404, 405, 410, 412, 415, 449–451, 461, 463 supertangent, 51, 53, 99, 101, 137, 145, 146, 161, 163, 243–245, 268, 270, 301, 303, 321, 323, 356, 357, 398, 399, 404, 405, 411, 412, 415, 450, 451, 461, 463

483

supertrigonometric, 113, 114, 461 functions, 49, 50, 53–55, 57, 97, 101, 102, 104, 105, 145, 146, 148, 242, 243, 245, 246, 267–272, 300, 303, 320, 323, 356, 357, 359, 361, 398, 399, 403, 405, 410, 412, 449, 451, 463 tangent, 148 I Incomplete beta function, 11 Euler, 9 gamma function, 5, 6, 284 Euler, 1 Integral Barnes contour, 144 Euler of first kind, 1 of the second kind, 10 representations, 11, 26, 81, 169, 189, 197, 205, 208, 209, 211, 217, 221, 227, 229, 232, 237, 246–248, 254, 257, 267, 272, 299, 310, 314, 352, 361, 362, 374, 385 transforms, 148, 149, 153 J Jacobi polynomial, 293, 297–307, 447 series, 294, 309 Jacobi theorem, 297 Jacobi–Luke polynomial, 309, 310, 312, 317, 320, 321, 323, 330–332, 343 K Koshliakov theorem, 142 Kummer confluent hypergeometric series, 284, 291 of first type, 237, 242–246, 254–257, 284 of second type, 265, 267–272, 276–280 L Laguerre differential equation, 346 function, 372 polynomial, 345–347, 352, 356, 357, 359, 361, 362, 367, 368, 373, 385, 390, 443 polynomial series, 345

484

Index

Laplace transform, 47, 57, 58, 73, 81, 144, 148, 149, 153, 164, 165, 169, 237, 242, 243, 252–254, 262, 264, 317, 318, 325, 327, 338, 339, 341, 345, 353, 362, 364, 385, 386, 388, 436, 440, 458 Legendre duplication formula, 4 polynomial, 445, 447–451, 453–456, 461, 466, 470 Logarithmic derivative, 11 Loop contour, 11 Luke generating function, 310 polynomial, 316 theorem, 293 M Mellin transform, 47, 144, 145, 148, 149, 153, 237, 242 N Numerator parameters, 97–102, 104–109 P Parameters denominator, 97–102, 104–109 numerator, 97–102, 104–109 Pearson theorem, 6 Pfaff theorem, 140 Pochhammer symbols, 6, 7, 12 Polynomial Jacobi, 293, 297–307, 447 Jacobi–Luke, 309, 310, 312, 317, 320, 321, 323, 330–332, 343 Laguerre, 345–347, 352, 356, 357, 359, 361, 362, 367, 368, 373, 385, 390, 443 series, 345 Legendre, 445, 447–451, 453–456, 461, 466, 470 Luke, 316 R Rainville function, 391, 398, 399, 401, 402 theorem, 295, 296, 391

Rodrigues formula, 294, 445, 459 S Schläfli’s integral formula, 446 Secant hypergeometric superhyperbolic, 75, 76, 107, 108, 137, 150, 151, 153, 162, 164, 255, 256, 279 Series Fourier–Jacobi, 309 Fourier–Legendre, 449 Hobson, 449 hypergeometric, 13, 14, 54, 79, 111, 142, 143, 180, 181, 185, 238, 256 Clausen, 13–18, 47, 49–55, 57, 73–75, 81, 97–100, 104–109, 136, 137 Fourier–Gauss, 147 Gauss, 13, 139, 142, 144–154, 169, 181, 233, 293 generalized, 13 Kummer confluent, 237, 242–246, 254–257, 265, 267–272, 276–280, 284, 291 polynomial Jacobi, 294, 309 Laguerre, 345 Sine hypergeometric, 147, 148, 243 superhyperbolic, 74, 76, 105, 108, 137, 149–153, 161, 163, 165, 169, 254, 256, 257, 279 Stirling theorem, 140 Supercosecant hypergeometric, 52, 53, 101, 146, 161, 163, 243–245, 269, 270, 302, 303, 321, 323, 357, 399, 405, 411, 412, 415, 450, 451, 462, 463 superhyperbolic, 277, 279, 306, 307, 331, 332, 368, 402, 408, 415, 454, 455, 466, 467 Supercosine, 148, 362, 385 hypergeometric, 51–53, 98, 101, 136, 145, 146, 149, 161, 163, 164, 169, 243–245, 267, 268, 270, 301, 303, 321, 323, 356,

Index

357, 398, 399, 404, 405, 410, 412, 415, 450, 451, 461, 463 superhyperbolic, 276, 278, 305, 307, 330, 332, 367, 368, 401, 402, 407, 408, 414, 453, 455, 466, 467 Supercotangent hypergeometric, 52, 53, 99, 101, 145, 146, 161, 163, 243–245, 268, 270, 301, 303, 321, 323, 357, 398, 399, 404, 405, 411, 412, 415, 450, 451, 462, 463 superhyperbolic, 277, 278, 305, 307, 331, 332, 368, 401, 402, 408, 414, 453, 455, 466, 467 Superhyperbolic functions, 113, 114 hypergeometric cosecant, 75, 76, 108, 151, 162, 164, 255, 256, 279 cosine, 74, 76, 106, 108, 137, 150–153, 162, 164, 165, 169, 254, 256, 257, 279 cotangent, 75, 76, 107, 108, 150, 151, 162, 164, 255, 256, 279 functions, 73, 74, 76, 81, 105, 108, 109, 149, 151, 153, 169, 254, 256, 257, 276, 277, 279, 280, 304, 307, 330, 332, 367, 368, 401, 402, 407, 408, 413, 415, 453, 455, 465, 467 secant, 75, 76, 107, 108, 137, 150, 151, 153, 162, 164, 255, 256, 279 sine, 74, 76, 105, 108, 137, 149–153, 161, 163, 165, 169, 254, 256, 257, 279 supercosecant, 277, 279, 306, 307, 331, 332, 368, 402, 408, 415, 454, 455, 466, 467 supercosine, 276, 278, 305, 307, 330, 332, 367, 368, 401, 402, 407, 408, 414, 453, 455, 466, 467 supercotangent, 277, 278, 305, 307, 331, 332, 368, 401, 402, 408, 414, 453, 455, 466, 467 supersecant, 277, 279, 306, 307, 331, 332, 368, 402, 408, 415, 454, 455, 466, 467 supersine, 276, 278, 305, 307, 330, 332, 367, 368, 385, 401, 402, 407, 408, 414, 453, 455, 465, 467

485

supertangent, 276, 278, 305, 307, 330, 332, 368, 401, 402, 407, 408, 414, 453, 455, 466, 467 tangent, 74, 76, 107, 108, 137, 150–152, 162, 164, 255, 256, 279 Supersecant hypergeometric, 52, 53, 99–101, 137, 146, 148, 161, 163, 243–245, 268, 270, 301, 303, 321, 323, 357, 399, 405, 411, 412, 415, 450, 451, 462, 463 superhyperbolic, 277, 279, 306, 307, 331, 332, 368, 402, 408, 415, 454, 455, 466, 467 Supersine hypergeometric, 50, 53, 97, 101, 136, 145, 146, 148, 149, 160, 162, 164, 169, 243–245, 267–270, 301, 303, 320, 323, 356, 357, 362, 398, 399, 404, 405, 410, 412, 415, 449–451, 461, 463 superhyperbolic, 276, 278, 305, 307, 330, 332, 367, 368, 385, 401, 402, 407, 408, 414, 453, 455, 465, 467 Supertangent hypergeometric, 51, 53, 99, 101, 137, 145, 146, 161, 163, 243–245, 268, 270, 301, 303, 321, 323, 356, 357, 398, 399, 404, 405, 411, 412, 415, 450, 451, 461, 463 superhyperbolic, 276, 278, 305, 307, 330, 332, 368, 401, 402, 407, 408, 414, 453, 455, 466, 467 Supertrigonometric functions, 54, 55, 104, 105, 147 hypergeometric functions, 49, 50, 53–55, 57, 97, 101, 102, 104, 105, 145, 146, 148, 242, 243, 245, 246, 267–272, 300, 303, 320, 323, 356, 357, 359, 361, 398, 399, 403, 405, 410, 412, 449, 451, 463 Szegö function of first type, 390, 403–405, 407, 408 of second type, 391, 410–415 notation, 293 theorem, 293, 390, 391

486

Index

T Tangent hypergeometric, 148 superhyperbolic, 74, 76, 107, 108, 137, 150–152, 162, 164, 255, 256, 279 Theorem Bateman, 142, 296 Brafman, 347, 392 Carlitz, 298 Catalan, 447 Clare, 447 Euler, 2, 139 limit, 7 Feldheim, 349 Gauss, 140 multiplication, 4 Glaisher, 446 Hankel integral, 11 Hille, 347 Hobson, 448 Jacobi, 297 Koshliakov, 142 Luke, 293

Pearson, 6 Pfaff, 140 Rainville, 295, 296, 391 Stirling, 140 Szegö, 293, 390, 391 Weierstrass, 4 Weisner, 349 Whittaker, 3 Winckler, 2 Transform Laplace, 47, 57, 58, 73, 81, 144, 148, 149, 153, 164, 165, 169, 237, 242, 243, 252–254, 262, 264, 317, 318, 325, 327, 338, 339, 341, 345, 353, 362, 364, 385, 386, 388, 436, 440, 458 Mellin, 47, 144, 145, 148, 149, 153, 237, 242 W Weierstrass product, 1, 2 Weierstrass theorem, 4 Weisner theorem, 349 Whittaker theorem, 3 Winckler theorem, 2