Padé Methods for Painlevé Equations (SpringerBriefs in Mathematical Physics) 9811629978, 9789811629976

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SPRINGER BRIEFS IN MATHEMATICAL PHYSICS 42

Hidehito Nagao Yasuhiko Yamada

Padé Methods for Painlevé Equations

SpringerBriefs in Mathematical Physics Volume 42

Series Editors Nathanaël Berestycki, University of Vienna, Vienna, Austria Mihalis Dafermos, Mathematics Department, Princeton University, Princeton, NJ, USA Atsuo Kuniba, Institute of Physics, The University of Tokyo, Tokyo, Japan Matilde Marcolli, Department of Mathematics, University of Toronto, Toronto, Canada Bruno Nachtergaele, Department of Mathematics, Davis, CA, USA Hal Tasaki, Department of Physics, Gakushuin University, Tokyo, Japan

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Nathanaël Berestycki (University of Cambridge, UK) Mihalis Dafermos (University of Cambridge, UK / Princeton University, US) Atsuo Kuniba (University of Tokyo, Japan) Matilde Marcolli (CALTECH, US) Bruno Nachtergaele (UC Davis, US) Hal Tasaki (Gakushuin University, Japan) 50 – 125 published pages, including all tables, figures, and references Softcover binding Copyright to remain in author’s name Versions in print, eBook, and MyCopy

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Hidehtio Nagao · Yasuhiko Yamada

Padé Methods for Painlevé Equations

Hidehtio Nagao Natural Sciences Division National Institute of Technology Akashi College Akashi, Japan

Yasuhiko Yamada Department of Mathematics Kobe University Kobe, Japan

ISSN 2197-1757 ISSN 2197-1765 (electronic) SpringerBriefs in Mathematical Physics ISBN 978-981-16-2997-6 ISBN 978-981-16-2998-3 (eBook) https://doi.org/10.1007/978-981-16-2998-3 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

The isomonodromic deformation equations such as Painlevé and Garnier systems are a very important class of nonlinear differential equations in mathematics and mathematical physics. In parallel with the numerous breakthroughs in discrete integrable systems (see a recent well-organized text book [14]), the study of the isomonodromic systems has also made much progress, particularly in the discrete (difference) cases. Today, we can access isomonodromic equations from various approaches: Painlevé test/Painlevé property, reduction from integrable hierarchies, Lax formulation, theory of orthogonal polynomials, affine Weyl group symmetries, algebraic geometry, cluster algebras, and so on. Among them, the Padé method we will explain in this monograph provides a very simple approach in both continuous and discrete cases. For a given function ψ(x), the Padé approximation (or interpolation) [2] supplies P(x) . The basic idea of polynomials P(x), Q(x) as approximants such as ψ(x) ∼ Q(x) the Padé method is to study the linear differential (or difference) equations satisfied by P(x) and ψ(x)Q(x). Choosing the approximation problem suitably, the linear equations give the Lax pair for some isomonodromic equations. In fact, the close relation between the isomonodromic equations and Padé approximations has been known classically, and the main advantage of this method is its simplicity. Once we set up a suitable Padé approximation/interpolation problem, we can obtain various explicit and exact (not approximate) results on the corresponding isomonodromic equations together with their Lax formulations and special solutions. Thus the method offers a very effective and instructive approach to continuous/discrete isomonodromic equations. The aim of this monograph is to explore a systematic and pedagogical study of the Padé method for both the continuous and discrete cases. In Chap. 1, we explain how the interesting special equations and special functions arise from the Padé problem through a toy example. In the process we will naturally encounter several key concepts on linear differential equations. In Chap. 2, we study the sixth Painlevé equation PVI (the most generic one among the six classical Painlevé equations) through a certain Padé approximation. Again, all the fundamental concepts and equations will arise naturally from the Padé problem. There exists a very nice q-difference analog of v

vi

Preface

the PVI equation: the q-PVI by Jimbo and Sakai. In Chap. 3, we study certain Padé approximation problems related to q-PVI together with its multivariable generalization (the q-Garnier system). The Padé interpolations are a more natural setting in which to approach the discrete Painlevé equations, and there are several kinds of Padé interpolation depending on the “grids” (the choice of the interpolating points: xs ). In Chap. 4, we study the q-Painlevé/Garnier systems using the Padé interpolation on a q-grid: xs = q s . In Chap. 5, the Padé interpolation on a q-quadraticgrid: xs = q s + cq −s is studied, and the result is related to a version of q-Garnier systems. Finally, Chap. 6 is devoted to an introduction to certain multicomponent Padé approximations/interpolations. As mentioned above, the relation between the isomonodromic equations and Padé approximations has been known classically. However, its systematic exploration including discrete cases was started rather recently. Many colleagues have contributed to make the project come to life, through their excellent work and valuable suggestions. We would in particular like to thank (in alphabetical order) Yusuke Ikawa, Kenji Kajiwara, Toshiyuki Mano, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta, Kanam Park, V. P. Spiridonov, Takao Suzuki, Teruhisa Tsuda, Satoshi Tsujimoto, and A. S. Zhedanov. The contents of this book grew out of the series of lectures by the second-named author at the University of Sydney in 2016. He sincerely thanks Prof. Nalini Joshi for providing such a nice opportunity, and the audiences for their interest. We would like to thank the referees for carefully reading the manuscript and giving valuable comments and suggestions. Last but not least, we sincerely thank Mr. M. Nakamura of Springer Japan for his patient encouragement. Akashi, Japan Kobe, Japan

Hidehito Nagao Yasuhiko Yamada

Acknowledgements H. Nagao is supported by JSPS KAKENHI Grant Number 19K14579, and Y. Yamada is supported by JSPS KAKENHI Grant Number 26287018.

Contents

1 Padé Approximation and Differential Equation . . . . . . . . . . . . . . . . . . . . 1.1 Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A Toy Example of Padé Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Contiguity Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Explicit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 6 7

2 Padé Approximation for P VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Derivation of the Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Deformation Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Explicit Solutions by Schur Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Extension to Garnier System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 More on the Schur Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 9 12 14 18 20 25

3 Padé Approximation for q-Painlevé/Garnier Equations . . . . . . . . . . . . . 3.1 Lax Pair for the q-Garnier Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The L 1 Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Special Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Relation to 2 × 2 Lax Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 37 40 45

4 Padé Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Cauchy–Jacobi Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Application to q-Garnier System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Special Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 A Duality Between q-Appell–Lauricella and q-HG Series . . . . . . . . .

49 49 51 55 57

5 Padé Interpolation on q-Quadratic Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Contiguous Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Lax Pair and the Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Special Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 65 68

vii

viii

Contents

6 Multicomponent Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Padé Approximations With Multicomponent . . . . . . . . . . . . . . . . . . . . 6.2 Application to the N -Garnier System . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Discrete Mahler Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 77 80

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Chapter 1

Padé Approximation and Differential Equation

P(x) Abstract The Padé approximation gives rational functions Q(x) which approximate a given function ψ(x). From the approximants P(x), Q(x), one can obtain interesting linear differential equations. We explain the derivation of such equations in a simple example.

1.1 Linear Differential Equations Consider a 2nd-order linear differential equation1 for an unknown function y = y(x): y  + a(x)y  + b(x)y = 0. ( =

d ) dx

(1.1)

It is enough to get two linearly independent solutions (fundamental solutions) to give a general solution; however, finding the fundamental solutions is not so easy in general. Conversely, the construction of the equation from its fundamental solutions is rather easy. Let y1 , y2 be fundamental solutions, then the differential equation can be written as the Wronskian determinant    y y  y     (1.2) W (y, y1 , y2 ) =  y1 y1 y1  = 0.  y2 y  y   2

2

Example If we know the fundamental solutions y1 , y2 of the form

1

Here we will summarize briefly some necessary facts on the linear differential equations. For more explanations, see e.g. [17, 67]. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd 2021 H. Nagao and Y. Yamada, Padé Methods for Painlevé Equations, SpringerBriefs in Mathematical Physics, https://doi.org/10.1007/978-981-16-2998-3_1

1

2

1 Padé Approximation and Differential Equation

yi = (x − c)ρi {1 + O(x − c)} (ρ1 = ρ2 ), x → c

(1.3)

then the corresponding differential equation takes the form y  +

 1 − ρ − ρ  ρρ 1  1 2 1 2 + O(1) y  + ) y = 0. + O( 2 x −c (x − c) x −c

(1.4)

The form of the solutions (1.3) or the form of the Eq. (1.4) can be used as an easy definition that x = c is a regular singular point with exponents ρ1 , ρ2 . In the case of c = ∞, the point x = ∞ is regular singular if the point w = 0 is regular singular under the change of variable x = w−1 . Explicitly it can be characterized by  1  yi = x −ρi 1 + O( ) , x → ∞ x or y  +

 1 + ρ + ρ ρ ρ 1  1 2 1 2 + O(1) y  + ) y = 0. + O( x x2 x

(1.5)

(1.6)

It is known that when the coefficients a(x), b(x) in (1.1) are holomorphic at some point, then the solutions are also holomorphic there. In other words, the singular points of the solutions are completely determined by the singularities of the coefficients a(x), b(x).2 If all the singular points are regular singular, such an equation is called Fuchsian. The general form of the 2nd-order Fuchsian differential equation with N regular singular points u 1 , . . . , u N −1 , u N = ∞ is given by (1.1) with a(x) =

N −1  i=1

N −1   bi ai ci  . , b(x) = + x − ui (x − u i )2 x − ui i=1

(1.7)

The exponents (αi , βi ) at each singular point u i are the basic data of the Fuchsian differential equation. It is represented by the following table called the Riemann scheme: x = u 1 u 2 · · · u N −1 ∞ (1.8) α1 α2 · · · α N −1 α N . β1 β2 · · · β N −1 β N Since ai = 1 − αi − βi and

2

 N −1 i=1

(1 − αi − βi ) = 1 + α N + β N , we have

This is a crucial property of linear differential equations. For the nonlinear equations, the solution may be singular even if the coefficients of the equation are holomorphic.

1.1 Linear Differential Equations

3 N  (αi + βi ) = N − 2.

(1.9)

i=1

This kind of relation among the exponents is called the Fuchs relation. We remark that when αi − βi ∈ Z a certain special situation may occur (see “Nonlogarithmic condition” in Sect. 2.1).

1.2 A Toy Example of Padé Approximation The Padé approximations are approximations of a given function by a rational function. A typical setting is as follows. For a given function ψ(x) holomorphic around x = 0, find polynomials P(x), Q(x) of degree m, n respectively, such that P(x) + O(x m+n+1 ). Q(x)

ψ(x) =

(1.10)

P(x) Namely, the ith derivative of ψ(x) − Q(x) , or equivalently those of P(x) − ψ(x)Q(x), vanishes at x = 0 for i = 0, 1, . . . , m + n.

The condition (1.10) gives m + n + 1 linear equations for the (m + 1) + (n + 1) coefficients of the polynomials P(x), Q(x), and hence, it generically determines the polynomials up to a common normalization factor. In the following, we will see that the Padé approximations give interesting differential (or difference) equations. As a toy example, we consider the case ψ(x) = (1 − x)α ,

(1.11)

where α is a complex parameter.

Proposition 1.1 The 2nd-order linear differential equation for y, y  , y  satisfied by y = P(x) and y = ψ(x)Q(x) is explicitly given by y  −

m + n x

+

α − 1   m(n + α) y + y = 0. x −1 x(x − 1)

(1.12)

4

1 Padé Approximation and Differential Equation

Proof The equation is obtained as the following Wronskian:    y y  y            u u u  = |u, u |y − |u, u |y + |u , u |y = 0,

(1.13)



P where u = . The computation of the coefficients goes as follows. ψQ Computation of |u, u |. Noting that −α ψ = , ψ 1−x

(1.14)

we have    P   |u, u | = ψ   Q 

   [m]   x    =ψ ψ    x [n] Q+Q   ψ P

x [m−1] x [n] 1−x

   x [m+n]  , =ψ  1−x 

(1.15)

where x [k] stands for a polynomial in x of degree at most k. On the other hand, due to the condition (1.10), we see that        P P  = O(x m+n ) (x → 0). (1.16) |u, u | =    ψ Q − P (ψ Q − P)  Hence, we have |u, u | = ψ

C x m+n = C x m+n (1 − x)α−1 , 1−x

(1.17)

where C is a constant. Computation of |u, u |. Taking a derivative of (1.17), we have 

 

|u, u | = |u, u | = C x

m+n

(1 − x)

α−1



m+n 1−α + . x 1−x

(1.18)

Computation of |u , u |. Using the relation α(α − 1) ψ  = , ψ (1 − x)2 we have

(1.19)

1.2 A Toy Example of Padé Approximation

   P P     |u , u | = ψ  ψ  ψ ψ    Q + 2 Q  + Q   ψ Q+Q ψ ψ    [m−1]   x x [m−2]  x [m+n−1]  = ψ  x [n] = ψ . [n]  x   (1 − x)2   1 − x (1 − x)2

5

      

(1.20)

(1.21)

Again, due to the condition (1.10), this is divisible by x m+n−1 . Hence |u , u | = C  x m+n−1 (1 − x)α−2 ,

(1.22)

where C  is another constant. Combining (1.17), (1.18), (1.22), we obtain y  −



α−1  m+n d + y + y = 0. x x −1 x(x − 1)

(1.23)

Finally, the constant d = C  /C can be determined as d = m(n + α) since the polynomial P(x) of degree m is a solution.  Simple derivation from singularity data One can derive the Eq. (1.12) from the Riemann scheme x =0 1 ∞ 0 0 −m . m + n + 1 α −α − n

(1.24)

Moreover, this data can be obtained directly from the singularity structure of the solutions P(x), ψ Q(x). Note that the exponents at x = 0 are 0 and m + n + 1 since we have fundamental solutions P(x) = O(x 0 ) and P(x) − ψ(x)Q(x) = O(x m+n+1 ) around x = 0.

From the example discussed above, we observe the following lessons. (i) The differential equation can be computed without knowing the explicit form of the polynomials P(x), Q(x). In fact, one can directly obtain the differential equation only by looking at the singularity structure of the solutions P(x), ψ(x)Q(x). (ii) Regardless of the degrees of polynomials P(x), Q(x), the coefficients of the differential equation are given by rational functions of small degree. This is because the Wronskian (1.13) is divisible by x m+n−1 thanks to the approximation condition

6

1 Padé Approximation and Differential Equation

(1.10). These are general features of the differential equations arising from the Padé approximation.

1.3 Contiguity Relations Besides the variable x, the polynomials P(x), Q(x) depend on the other parameters α, m, n. Here we will consider the discrete deformation w.r.t. the parameter α. Proposition 1.2 For the shift T ± : α → α ± 1, we have the differential/difference equations satisfied by y = P(x) and y = ψ(x)Q(x) as follows:

and

x y  + (α + n)T −1 (y) − (α + n)y = 0,

(1.25)

(mx − m + α)y + (m − α)T (y) + x(1 − x)y  = 0.

(1.26)

Proof These equations can be obtained from    y y  T ± (y)   ± ±   ±    u u T ± (u)  = |u, u |T (y) − |u, T (u)|y + |u , T (u)|y = 0,

(1.27)



P as before. ψQ Using the relation

where u =

T −1 (ψ(x)) = (1 − x)α−1 = (1 − x)−1 ψ(x),

(1.28)

and counting the degree and vanishing order in x at x = 0 and x = ∞, we have ψ(x) C1 x m+n+1 , 1−x ψ(x) |u , T −1 (u)| = C2 x m+n , 1−x

|u, T −1 (u)| =

(1.29) (1.30)

where C1 , C2 are some constants. Together with (1.17), we have C T −1 (y) − C2 x y  + C2 y = 0.

(1.31)

The ratios of constants C, C1 , C2 can be determined by the condition that the equation has solutions of the form y = P = 1 + O(x) (x → 0) and y = ψ Q = O(x α+m ) (x → ∞), as in (1.25). Similarly, from T (ψ(x)) = (1 − x)ψ(x), we see

1.3 Contiguity Relations

7

|u, T (u)| = ψ(x)C3 x m+n+1 , |u , T (u)| =

ψ(x) (C4 x + C5 )x m+n . 1−x

(1.32)

Hence we have (C4 x + C5 )y − C T (y) + C3 x(1 − x)y  = 0.

(1.33)

Again, the ratios of constants C, C3 , C4 , C5 can be determined from the existence of solutions of the form y = 1 + O(x) (x → 0), y = O(x m ) (x → ∞) and y =  O((1 − x)α ) (x → 1), and we have (1.26). Contiguity relations Equations (1.25), (1.26) describing the α-parameter dependence are called contiguity relations. Eliminating T (y), T −1 (y) from these two equations, one can recover the differential equation (1.23). Furthermore, eliminating y  , one can also derive the following difference equation w.r.t. α:   (n + α)(x − 1)T −1 (y) + (m − n)(x − 1) + (2 − x)α y + (m − α)T (y) = 0. (1.34) In this sense, the contiguity relations (1.25), (1.26) are more fundamental than (1.23) and (1.34).

1.4 Explicit Solutions Though we do not need the explicit form of P(x), Q(x) to derive the differential (and contiguity) equations, the information about the explicit form also plays an important role in our story. In the case of the toy example, they can be determined by solving the differential equation (1.12), and given as P(x) = 2 F1 (−m, −α − n, −m − n; x), Q(x) = 2 F1 (−n, α − m, −m − n; x),

(1.35)

where 2 F1 is the Gauss hypergeometric function 2 F1 (a, b, c; x)

=1+

a(a + 1)b(b + 1) 2 ab x+ x + ··· . c1 c(c + 1)2!

(1.36)

Note that 2 F1 (a, b, c; x) is polynomial (terminating) if a ∈ Z≤0 or b ∈ Z≤0 and c∈ / Z≤0 . The function 2 F1 (a, b, c; x) is a solution of the following equation:

8

1 Padé Approximation and Differential Equation

x(1 − x)y  + {c − (a + b + 1)x}y  − aby = 0. This equation (Gauss hypergeometric equation) scheme x =0 1 0 0 1−c c−a−b

(1.37)

is characterized by the Riemann ∞ a . b

(1.38)

The Eq. (1.12) is a special case of this with two exponents in Z. Derivation from Euler transformation The Padé approximation for ψ(x) = (1 − x)α can be obtained from the Euler transformation formula: (1 − x)a+b−c 2 F1 (a, b, c; x) = 2 F1 (c − a, c − b, c; x),

(1.39)

which follows by noting the change of exponents (1 − x)a+b−c ×

x =0 0 1−c

1 0 c−a−b

∞ x =0 a = 0 b 1−c

1 0 −c + a + b

∞ c−a , c−b (1.40)

and consider the holomorphic solution around x = 0 which is generically unique up to a normalization. Naively, a specialization (a, b, c) = (−m, −α − n, −m − n) of the formula (1.39) seems to give the approximation relation (1.10), but the error terms are missing. The correct result can be obtained by putting (a, b, c) = (−m, −α − (n + ), −m − (n + )) and taking the limit  → 0. This is exactly the method H. Padé used in [48] to obtain the approximation of the power function (1 − x)α as ratios of the terminating hypergeometric functions. In a similar way, starting from the function ψ(x) = ex , one obtains the differential equation (1.41) x y  − (x + m + n)y  + my = 0. The corresponding Padé interpolants are P(x) = 1 F1 (−m, −m − n; x) and Q(x) = − n, x), where 1 F1 is the confluent hypergeometric function

1 F1 (−n, −m

1 F1 (α, γ ; x)

=1+

α(α + 1) 2 α x+ x + ··· . γ γ (γ + 1)2!

(1.42)

This result can be obtained from the case ψ(x) = (1 − x)α by changing the variable x → x/α and taking a limit α → ∞.

Chapter 2

Padé Approximation for PVI

Abstract We will consider the Padé approximation problem associated with the function ψ(x) = (1 − x)α (1 − xt )β , and derive a pair of linear differential equations from it. Since the compatibility condition of the pair is the sixth Painlevé equation PVI , we obtain special solutions of the PVI equation from the Padé problem.

2.1 Derivation of the Differential Equation In this chapter, we consider the following Padé approximation problem. Find polynomials P(x), Q(x) of degree m, n such that ψ(x) = where

P(x) + O(x m+n+1 ), Q(x)

ψ(x) = (1 − x)α (1 −

x β ) . t

(2.1)

(2.2)

We will see that this problem is related to the sixth Painlevé equation PVI . First, in this section, we show the following: Theorem 2.1 [63] The differential equation satisfied by P(x) and ψ(x)Q(x) is given explicitly as y  + a(x)y  + b(x)y = 0,   α−1 β −1 1 m+n + + + , a(x) = − x x −1 x −t x −q   q(q − 1) p t (t − 1)H 1 m(α + β + n) + − , b(x) = x(x − 1) x −q x −t

(2.3) (2.4) (2.5)

where q, p, H are certain parameters. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd 2021 H. Nagao and Y. Yamada, Padé Methods for Painlevé Equations, SpringerBriefs in Mathematical Physics, https://doi.org/10.1007/978-981-16-2998-3_2

9

10

2 Padé Approximation for PVI

Later, the parameter H will be determined as a function of q, p, see (2.20). The parameters q, p are called an accessory parameters since they cannot be determined from the prescribed singularity data (at x = 0, 1, t, ∞). Such parameters will play the role of unknown variables of the Painlevé equation. Proof As before, the equation can obtained from the Wronskian      y y  y   P          u u u  = |u, u |y − |u, u |y + |u , u |y = 0, u = ψ Q . Computation of a(x). Noting that

ψ ψ

=

α x−1

+

β , x−t

(2.6)

we have

 [m]    x P   x [m−1] P [m+n+1]     [n+1] =ψ =ψ x x . |u, u | = ψ  ψ   [n]   (x − 1)(x − t)  x (x − 1)(x − t)  Q ψ Q+ Q  (2.7) The notation x [k] stands for a polynomial in x of degree at most k. Since this determinant is divisible by x m+n , we have |u, u | = ψ x m+n

C(x − q) , (x − 1)(x − t)

(2.8)

where C, q are constants with respect to x. Accordingly, we have − a(x) =

|u, u | α−1 β −1 1 m+n = {log(|u, u |)} = + + + . (2.9)  |u, u | x x −1 x −t x −q

Computation of b(x). Since ψ  =

x [2] ψ, (x−1)2 (x−t)2

|u , u | = ψ x m+n−1 and hence b(x) =

we have

x [2] , (x − 1)2 (x − t)2

C0 + C1 x + C2 x 2 |u , u | = .  |u, u | x(x − 1)(x − t)(x − q)

(2.10)

(2.11)

From an existence of a polynomial solution P of degree m, the constant C2 is determined as C2 = m(α + β + n). It is convenient to parametrize the expression (2.11) by the residues p and −H at x = q and x = t respectively, then b(x) takes the desired form (2.5). The parameter H will be determined as a function of p, q (see Proposition 2.1).  The Riemann scheme of (2.3) is

2.1 Derivation of the Differential Equation

x =0 0 m+n+1

1 0 α

11

t 0 β

q 0 2

∞ . −m −α − β − n

(2.12)

This can be seen directly from the singularity structure of the solutions P(x), ψ(x)Q(x). The singularity at x = q is called an apparent singularity since the solutions are regular there. Note that the exponents at x = q are (0,2) while the exponents at ordinary regular points are (0,1). In this example such an apparent singular point is needed to fulfill the Fuchs relation. Non-logarithmic condition We make a brief remark on the case where the exponents have an integer difference. A 2nd-order equation (1.1) with regular singular point at x = q with exponents ρ1 , ρ2 is expected to have two linearly independent power series solutions yi = (x − q)ρi

∞ 

ci,n (x − q)n (i = 1, 2)

(2.13)

n=0

around x = q. This is true for generic exponents ρ1 , ρ2 where the coefficients {c1,n } and {c2,n } can be determined through the linear recursion relations separately. However, we need some additional considerations if ρ1 − ρ2 ∈ Z. We assume ρ2 = ρ1 + k (k ∈ Z≥0 ). There is no problem for the solution y2 . However, for the solution y1 , there may be a trouble since the recursion relation for {c1,n } will get mixed up with that for {c2,n }. In this case, the recursion relation for {c1,n } has the form (2.14) n(n − k)c1,n = Rn (n > 0), where Rn is a certain linear function of c1,0 , c1,1 , . . . , c1,n−1 . We have two cases. (Case i) Rk = 0. One can solve for {c1,n } with two free parameters c1,0 and c1,k . In this case, we still have two linear independent solutions in the power series form. (Case ii) Rk = 0. There is no solution for c1,k . This means that there is no power series solution y1 in (2.13). Hence, the ansatz for y1 in (2.13) is not available and we replace it by including a logarithmic term1 y1 = (x − α)ρ1

∞ 

c1,n (x − α)n + Const.y2 log(x − α).

(2.15)

n=0

(Case i) occurs only when the coefficients of the differential equations satisfy a special relation (the non-logarithmic condition). For instance, if x = q is the apparent singularity with exponents (ρ1 , ρ2 ) = (0, 2), we should have two holomorphic solutions of the form 1

See [67] Sect. 10.32. A convenient way to obtain such a solution is known as the Frobenius method: consider a regularized “solution” for ρ = ρ2 +  and expanding (or taking the derivative) w.r.t .

12

2 Padé Approximation for PVI

y1 = 1 + O(z),

y2 = z 2 (1 + O(z)),

(2.16)

where z = x − q. This means that the coefficient c2 of the series solution y1 = 1 + c1 z + c2 z 2 + · · · ,

(2.17)

should be a free parameter. As an example, let us put the expansion coefficients of a(x), b(x) around z = 0 as a(x) = z −1 (−1 + a1 z + · · · ), b(x) = z −2 (0 + b1 z + b2 z 2 + · · · ).

(2.18)

Putting these into (2.3), we see c1 = b1 , and c2 is a free parameter if and only if a1 b1 + b2 + b1 2 = 0.

(2.19)

This is the non-logarithmic condition in the case of exponents (ρ1 , ρ2 ) = (0, 2). For the differential equation (2.3) arising from the Padé problem, this condition should be satisfied since the solutions P(x), ψ(x)Q(x) are non-logarithmic at x = q. Proposition 2.1 From the non-logarithmic condition at x = q, the parameter H in (2.5) is determined as H=

q(q − 1)(q − t) t (t − 1)

 p2 −

m + n + 1 q

+

 α β − 1 m(α + β + n) + p+ . q −1 q −t q(q − 1)

(2.20)

Proof From (2.5), we have expansions (2.18) where α−1 β −1 m+n − − , b1 = p, q q −1 q −t m(α + β + n) (2q − 1) p t (t − 1)H + − , b2 = − q(q − 1)(q − t) q(q − 1) q(q − 1) a1 = −

hence from (2.19) we obtain (2.20).

(2.21)



2.2 Deformation Equation In the same situation as the previous section, we compute another linear equation between y, y  , yt = ∂∂ty satisfied by P(x) and ψ(x)Q(x). Here, to fix the normalization of the polynomials P(x), Q(x), we put P(0) = 1 (hence Q(0) = 1). Proposition 2.2 We have

2.2 Deformation Equation

13

t (t − 1)(x − q) yt + x(x − 1)y  − (q − 1) px y = 0. t −q

(2.22)

Proof The equation is given by the following determinant:    y y  yt        u u ut  = |u, u |yt − |u, ut |y + |u , ut |y = 0.

(2.23)

The determinant |u, u | is already computed in (2.8), and we will compute the remaining two. βx , we have Since ψψt = t (t−x)    P  |u, ut | = ψ   Q 

   [m]   x    =ψ ψt  x [n]  Q + Qt   ψ Pt

x [m] x [n+1] x −t

   x [m+n+1]  . =ψ  x −t 

(2.24)

Since this is divisible by x m+n+1 , we have |u, ut | = c1 ψ

x m+n+1 , x −t

where c1 is a constant. Next, for the determinant |u , ut |, we have    P Pt   |u , ut | = ψ  ψ  ψ t   Q + Qt  ψ Q+Q ψ    x [m−1] x [m]  =ψ x [n+1] x [n+1]   (x − 1)(x − t) x − t

          x [m+n+1]  . =ψ  (x − 1)(x − t) 

(2.25)

(2.26)

Due to the normalization P(0) = 1, |u , ut | is divisible by x m+n+1 , and we have |u , ut | = c2 ψ

x m+n+1 , (x − 1)(x − t)

(2.27)

where c2 is a constant. Thus we have the following relation: C(x − q)yt + c1 x(x − 1)y  + c2 x y = 0.

(2.28)

The ratios of the constants C, c1 , c2 are determined by the following conditions. (i) From (2.3), we see that y  = py for x = q. (ii) From the regularity of P(x), Q(x) at x = t, we have

14

2 Padé Approximation for PVI

lim

x→t

|u, ut | ψt = lim = 1.  x→t ψx |u, u |

(2.29) 

As a result, we have the desired Eq. (2.22).

2.3 Explicit Solutions by Schur Functions From the solution of the Padé problem (2.1), the parameters q (and p also) in Eqs. (2.3), (2.22) can be determined as a function of t. By construction, these equations are compatible for such q (and p). Since the compatibility condition is equivalent to the PVI equation (see Sect. 2.6), we see that the function q(t) determined by Padé problem gives a special solution for PVI . Here, we will carry out the computation of q(t). An explicit formula of the polynomials P(x), Q(x) is given by determinants (Schur functions). This formula is useful to obtain special solutions of various Painlevé type equations. We consider the Padé problem (2.1) for ψ(x) =

∞ 

pk x k ,

(2.30)

k=0

( p0 = 1, pi = 0, i < 0). We define the Schur function sλ corresponding to the sequence of integers λ = (λ1 , λ2 , · · · ) by the Jacobi–Trudi formula s(λ1 ,...,λl ) = det( pλi −i+ j )li, j=1 , e.g.

   p3 p4 p5    =  p2 p3 p4  .  0 1 p1 

s(3,3,1) =

(2.31)

(2.32)

Theorem 2.2 The polynomials P(x) and Q(x) are given by P(x) =

m 

s(m n ,i) x i , Q(x) =

i=0

n 

s((m+1)i ,m n−i ) (−x)i ,

i=0

where a sequence of integers (m, m, . . . , m) is abbreviated as m n .



n

(2.33)

2.3 Explicit Solutions by Schur Functions

15

Example In the case of (m, n) = (3, 2), we have P(x) =

+

Q(x) =

−

x+ x+

x2 +

x 3,

x 2.

(2.34) (2.35)

The normalization of P(x), Q(x) in (2.33) is such that P(0) = Q(0) = τm,n , where τm,n = sm n

(2.36)

is the Schur function corresponding to the rectangular Young diagram m n .

Proof of Theorem 2.2. Let us check that the polynomials P(x), Q(x) given by (2.33) satisfy the Padé approximation condition ψ(x)Q(x) − P(x) = O(x m+n+1 ).

(2.37)

The polynomial Q(x) can be written as   pm  ···   Q(x) =   ···   xn

pm+1 · · · pm pm+1 .. . ··· ··· ··· ···

 · · · pm+n  · · · · · ·   .. , .  pm pm+1  x 1 

(2.38)

Hence we have   pm   ··· ∞   ψ(x)Q(x) =   i=1  · · ·   pi−n =

m  i=0

as desired.

pm+1 · · · pm pm+1 .. . ··· ··· ··· ···

s(m n ,i) x i +

∞ 

 · · · pm+n  · · · · · ·  ∞  i  n i .. x = s(m ,i) x  .  i=0  pm pm+1  pi−1 pi 

(2.39)

s(m n ,i) x i = Pm (x) + O(x m+n+1 ),

i=m+1



Applying the formulas (2.33), we derive an explicit form of a special solution q(t) of PVI .

16

2 Padé Approximation for PVI

Theorem 2.3 The following function gives a special solution of PVI for m, n ∈ Z≥0 : q(t) =

t (m + n + 1) τm,n τm+1,n+1 . m − n − α − β τm+1,n τm,n+1

(2.40)

Proof To obtain the explicit form of the parameter q in (2.8), we put |u, u | = ψ(1 − x)−1 (1 − x/t)−1 x m+n (A + Bx).

(2.41)

From (2.39) we have ψ Q − P = vx m+n+1 + O(x m+n+2 ), where v = s(m n ,m+n+1) = (−1)n τm+1,n+1 .

(2.42)

Then we have     P P   |u, u | =     ψ Q − P (ψ Q − P)    P(0) P(1)  =  vx m+n+1 (m + n + 1)vx m+n 

(2.43)     + O(x m+n ) 

= (m + n + 1)P(0)vx m+n + O(x m+n+1 ). Hence A = (−1)n (m + n + 1)τm,n τm+1,n+1 . On the other hand, for x → ∞ we have    P   −1  −1  P ψ |u, u | = ψ    ψ Q (ψ Q)    τm,n+1 mτm,n+1  =  (−1)n τm+1,n (−1)n (α + β + n)τm+1,n and hence

q=−

(2.45)    m+n−1 + O(x m+n−2 ), x 

(α + β − m + n) τm+1,n τm,n+1 . t

(2.46)

t (m + n + 1) τm,n τm+1,n+1 B = , A m − n − α − β τm+1,n τm,n+1

(2.47)

B = (−1)n Thus we get

(2.44)

as desired. Remark 2.1 The function ψ(x) (2.2) has an expansion



2.3 Explicit Solutions by Schur Functions

ψ(x) = (1 − x)α (1 −

17

 (−α)i (−β)l x β ) = t −l x i+l , t i! l! i,l≥0

(2.48)

where (α)k = α(α + 1) · · · (α + k − 1). Hence the coefficient pk can be written as pk = =

k k  (−α)k−l (−β)l −l (−α)k  (−k)l (−β)l −l t = t (k − l)! l! k! l=0 (α − k + 1)l l! l=0

(2.49)

(−α)k −1 2 F1 (−k, −β, α − k + 1; t ). k!

Thus we obtained the explicit form of the special solution (2.40) of PVI given in terms of the determinants of Jacobi polynomials. Though such solutions are well known, the derivation [63] explained here is quite simple. τm,n as the τ -function The most fundamental object in the isomonodromic deformation is the τ -function [20–22]. It can be defined as dtd log τ = H (q(t), p(t), t) up to a normalization. More 2 −a 2 , the function precisely, for a solution p, q for the PVI Eq. (2.97) with k = (b+c+d) 4 σ defined by (2.50) σ = t (t − 1)H (q, p) + k1 t + k2 , 2

where k1 = d(d+2b+2c)−a , k2 = 4 called the σ -form [21, 45]:

a 2 −b2 +c2 −4bd−d 2 8

satisfies the following equation

σ  (t (t − 1)σ  )2 + (2σ  (tσ  − σ ) − (σ  )2 + c1 c2 c3 c4 )2 −

4 

(σ  − ci2 ) = 0,

i=1

(2.51) where  = dtd and (c1 , · · · , c4 ) = 21 (a + d, a − d, b + c, b − c). For the special solutions obtained from the Padé method, the determinants τm,n give the τ -function. In fact, one can check that the following function σ = t (t − 1)(log τm,n ) + (1 − t)c12 +

1  ci c j , 2 1≤i< j≤4

(2.52)

satisfies (2.51) with (c1 , . . . , c4 ) = ( −α+m−n , α−m−n , α+m+n , α−m+n + β). For the 2 2 2 2 derivation of similar and more general formulae, see [32] for example.

Remark 2.2 We have seen the connection between certain Padé approximations and isomonodromic equations. In fact, it has been known that there are close connections among the orthogonal polynomials, Padé approximations, and Painlevé equa-

18

2 Padé Approximation for PVI

tions (see [4, 28, 30, 31, 62] for example). Our main strategy is to formulate and study various differential/discrete isomonodromic equations starting suitable Padé problems. We will find this is a useful method since the Padé problem is easy to attack.

2.4 Extension to Garnier System Soon after the work by R. Fuchs, an extension of the PVI was obtained by Garnier [8] as the isomonodromic deformation of the Fuchsian equation on P1 with N + 3 regular singular points at {0, 1, t1 , . . . , t N , ∞} (together with N apparent singularities {q1 , . . . , q N }). This system can be written as a multi-time Hamiltonian system with 2N unknown variables pi , qi and N time variables ti (i = 1, . . . , N ), and called the N -Garnier system [8, 9] (see [17] for modern exposition). For convenience of the readers, we give a short introduction in Sect. 2.6. The Garnier system includes the PVI as the N = 1 case. To obtain the N -Garnier system, we consider the following Padé problem. Find polynomials P(x), Q(x) of degree m, n such that ψ(x) =

P(x) + O(x m+n+1 ), Q(x)

where ψ(x) = (1 − x)κ

N 

(1 −

i=1

(2.53)

x αi ) . ti

(2.54)

In this case, the Riemann scheme is given by x =0 0 m+n+1

1 0 κ

t1 0 α1

··· ··· ···

tN 0 αN

∞ −m N −n − κ − i=1 αi

λ1 0 2

··· ··· ···

λN 0 , (2.55) 2

where λ1 , . . . , λ N are apparent singularities. Coefficient a(x) of the differential equation L : y  + a(x)y  + b(x)y = 0 has the form N N κ − 1  αi − 1  1 m+n + + + . (2.56) − a(x) = x x − 1 i=1 x − ti x − λi i=1 Define variables μi and K i by

2.4 Extension to Garnier System

19

μi = Res b(x), x=λi

K i = − Res b(x).

(2.57)

x=ti

By the non-logarithmic condition at x = λi , K i is determined as a rational function of {λ j , μ j }. We also have the deformation equations of the form {∂t j + G j (x)∂x + F j (x)}y(x) = 0 ( j = 1, . . . , N ),

(2.58)

then, as the compatibility equations, we have (see Sect. 2.6) ∂Kj ∂λi = , ∂t j ∂μi

∂Kj ∂μi =− ∂t j ∂λi

(i, j = 1, 2, . . . , N ).

(2.59)

This multi-time Hamiltonian system is the N -Garnier system. The expansion of the function ψ(x) (2.54) is given by ψ(x) =

∞  (−κ)l l=0

(1)l



xl ·

x |m|

m i ≥0

N  (−αi )m i=1

where (a)k = a(a + 1) · · · (a + k − 1) and |m| = pk =

(1)m i

N i=1

i

ti −m i ,

(2.60)

m i . Hence we have

(−κ)k FD (−k, −α1 , . . . , −α N , κ − k + 1; t1−1 , . . . , t N−1 ), (1)k

(2.61)

where FD is the Appell–Lauricella function  (α)|m| FD (α, β1 , . . . , β N , γ ; x1 , . . . , x N ) = (γ )|m| m ≥0 i



N  (βi )m i=1

 i

(1)m i

xim i

.

(2.62)

In order to specify {λi } as functions of {t j } (or to give special solutions of the N Garnier systems), it is convenient to consider the polynomial F(x) = i=1 (x − λi ). Since F(x) appears as the non-trivial factor of the Wronskian |u, u |, it can be explicitly determined by the following special values at x = ti (i = 0, 1, . . . , N ): F(ti ) =

ai n

N

j (=i)=0 (ti − t j ) N − m + i=0 ai

(Tαi τm+1,n )(Tα−1 τm,n+1 ) i τm+1,n τm,n+1

,

(2.63)

where t0 = 1, α0 = κ and Tαi means the shift αi → αi + 1. We also have F(0) =

(m + n + 1) n−m+

N

j=0 (−t j ) τm+1,n τm,n+1

N

i=0

ai

τm+1,n τm,n+1

.

(2.64)

20

2 Padé Approximation for PVI

The Eq. (2.63) can be derived from P(ti ) = tim Tai τm,n+1 ,

Q(ti ) = (−ti )n Ta−1 τm+1,n , i

(2.65)

which follows from the formulae (2.69) in the next section. The corresponding τ -functions for these special solutions are given by the determinants τm,n (see for example [59] where more general results are discussed). We will study the N -Garnier system further in Sect. 6.2.

2.5 More on the Schur Functions We will discuss some further properties of Schur functions which play an important role in various areas of mathematical physics ([34] see also [60] for further generalization). For any partition λ = (λ1 , λ2 , . . . , λ ), the Schur functions sλ can be viewed as minor determinants of the infinite matrix X = ( p j−i )i, j∈Z : ⎡

.. .. ⎢ . . ⎢ p0 ⎢ ⎢ X =⎢ ⎢ ⎢ ⎢ ⎣ O

⎤ . ··· ⎥ ⎥ p1 p2 · · · ⎥ ⎥ p0 p1 p2 · · · ⎥, ⎥ p0 p1 p2 · · · ⎥ ⎥ p0 p1 p2 · · · ⎦ .. .. .. . . . ···

..

(2.66)

where the row and column indices are chosen as I = (1 − , 2 − , . . . , −1, 0) and J = λ − (0, 1, . . . , − 1) respectively. The index set J is conveniently represented by a figure called a Maya diagram. Example For λ = (5, 3, 3, 2, 0), we have I = (−4, −3, −2, −1, 0), J = (5, 2, 1, −1, −4), hence the minor determinant gives the Schur function sλ : i\ j −4 −1 1 2 5   p5 p6 −4 p0 p3 p5 p6 p9  p p −3 p2 p4 p5 p8 =  2 3 −2 p1 p3 p4 p7  p1 p2  p0 −1 p0 p2 p3 p6 0 p1 p2 p5

p7 p4 p3 p1

 p8  p5  = sλ . p4  p2 

(2.67)

2.5 More on the Schur Functions

21

In the first equality we used the fact that, for any N × N matrix A, a transformation A → J A T J , J = (δi+ j,N )i,N j=1 (i.e. the transposition along the anti-diagonal) preserves the determinant. Since sλ = s(λ,0) , one can consider that the partition λ = (λ1 , λ2 , . . . , 0, 0, · · · ) is of infinite length. Then the index set J = (5, 2, 1, −1, −4, −5, · · · ) and the corresponding Maya diagram is as follows: λ=

↔

· · · −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 · · · · · · w w w g g w g w w g g w g g g· · ·

We introduce operators which create/annihilate w’s. For a parameter x ∈ C, let V (x), V (x) be shift operators acting on a polynomial f in variables { pi } as ∗

  V ∗ (x) f = f 

pi →

i j=0

x − j pi− j

  , V (x) f = f 

pi → pi −x −1 pi−1

.

(2.68)

Proposition 2.3 The polynomials P(x) and Q(x) in (2.33) can be expressed as P(x) = x m V ∗ (x)τm,n+1 , Q(x) = (−x)n V (x)τm+1,n . Proof (i) Put pˆ i =

i j=0

(2.69)

x − j pi− j and consider the determinant

V ∗ (x)τm,n+1

  pˆ m   pˆ m−1  =    pˆ m−n

 pˆ m+1 · · · pˆ m+n  pˆ m · · · pˆ m+n−1  . .. ..  . .  · · · pˆ m−1 pˆ m 

(2.70)

By the relation pˆ i − x −1 pˆ i−1 = pi , the pˆ i ’s in the determinant can be restored to pi except for the last row. Hence we have V ∗ (x)τm,n+1 =

m 

s(m n ,m− j) x − j = x −m P(x).

(2.71)

i=0

(ii) Put pˆ i = pi − x −1 pi−1 and consider the determinant

V (x)τm+1,n

  pˆ m+1   pˆ m  =    pˆ m−n

 · · · pˆ m+n  · · · pˆ m+n−1  . ..  .  · · · pˆ m pˆ m+1 

pˆ m+2 pˆ m+1 .. .

(2.72)

22

2 Padé Approximation for PVI

Expanding pˆ i into (a) pi or (b) −x −1 pi−1 in each row, non-zero results arise only for the choices (a) . . . (a)(b) . . . (b) from top to bottom, since the term vanishes for the choice . . . (b)(a) . . .. Hence we have V (x)τm+1,n =

n 

s((m+1)i ,m n−i (−x)−(n−i)) = (−x)−n Q(x),

(2.73)

i=0



as desired. Proposition 2.4 For any partition λ = (λ1 ≥ λ2 ≥ . . .), we have V ∗ (x)sλ = ψ(x)

∞  (−1)i−1 s(λ1 +1,λ2 +1,...,λi−1 +1,λˇi ,λi+1 ,λi+2 ,...) x i−1−λi , (2.74) i=1

V (x)sλ = ψ(x)

−1



s(k,λ) x k ,

(2.75)

k∈Z

ˇ k, . . .) = (. . . , i, j, . . .) and (k, λ) = (k, λ1 , λ2 , . . .). where (. . . , i, j, Proof First consider (2.74). By the column operations using pˆ i − x −1 pˆ i−1 = pi , we have      pˆ λ1 pˆ λ1 +1 · · · pˆ λ1 + −1   pˆ λ1 pλ1 +1 · · · pλ1 + −1       pˆ λ2 −1 pˆ λ2 · · · pˆ λ2 + −2   pˆ λ2 −1 pλ2 · · · pλ2 + −2      ∗ V (x)sλ =  .  =  ..  (2.76) .. ..   .   .. . .      pˆ λ   p λ     −1  pˆ λ1 + i=1 pλ1 +i x i pλ1 +1 · · · pλ1 + −1     pˆ λ −1 + −1 pλ +i−1 x i−1 pλ · · · pλ + −2  2 2 2  2  i=1 = . .. ..   . .     p λ

We consider that the partition λ = (λ1 , λ2 , . . . , 0, 0, 0) is of large length , and take the limit → ∞, then we have   −λ  x 1 pλ1 +1 pλ1 +2 · · ·    −λ +1  x 2  −λ +2 pλ2 pλ2 +1 · · ·  ∗ (2.77) V (x)sλ → ψ(x)  x 3 pλ −1 pλ · · ·  3 3    .. . .   . . = ψ(x)

∞  (−1)i x i−1−λi sλ1 +1,λ2 +1,...,λi−1 +1,λi+1 ,λi+2 ,...) , i=1

hence the Eq. (2.74) is proved. For (2.75), we have

2.5 More on the Schur Functions

23

     pk pk+1 · · ·    1 x −1 · · ·            pλ1 −1 pλ1 · · ·  k   pλ1 −1 pλ1 · · · k x s(k,λ) x = = ψ(x)  ..    . .. ..  .    .. . . k∈Z k∈Z       p λ  p λ  = ψ(x){sλ − s(λ1 −1,λ2 ,...) x −1 + s(λ1 −1,λ2 −1,λ3 ,...) x −2 − · · · }

(2.78)

= ψ(x)V (x)sλ , 

hence we obtain (2.75).

Since Schur functions are minor determinants, there are various bilinear relations among them. Proposition 2.5 We have s(k,λ) .s(l−1,m,λ) + s(l,λ) .s(m−1,k,λ) + s(m,λ) .s(k−1,l,λ) = 0,

(2.79)

V (a)sλ .s(l−1,m,λ) − as(l,λ) .V (a)s(m,λ) + as(m,λ) .V (a)s(l,λ) = 0, bV (a)sλ .V (b)s(m,λ) − aV (b)sλ .V (a)s(m,λ) + (a − b)s(m,λ) .V (a)V (b)sλ = 0,

(2.80) (2.81)

(b − c)V (a)sλ .V (b)V (c)sλ + (c − a)V (b)sλ .V (c)V (a)sλ + (a − b)V (c)sλ .V (a)V (b)sλ = 0.

(2.82)

Proof Though these also can be proved using fermion operators, we will give more elementary derivations. The Eq. (2.79) follows from the identity (Plücker relation) d12 d34 + d23 d14 + d31 d24 = 0,

(2.83)

for determinants di j = |ai , a j , x1 , . . . , xn |, ai , xi ∈ Cn . The remaining  relations a k s(k,λ) = (2.80)–(2.82) can be derived from (2.79) using s(m−1,k,λ) = −s(k−1,m,λ) , ψ(a)V (a)sλ ,

 l∈Z

k∈Z

b V (a)s(l,λ) l

b = (1 − )ψ(b)V (a)V (b)sλ , etc. a



Proposition 2.6 In particular, for τm,n = s(m n ) , we have τm−1,n .V (a)τm+1,n + τm,n+1 .V (a)τm,n−1 − τm,n .V (a)τm,n = 0,

(2.84)

−1

τm−1,n .V (a)τm,n+1 − τm,n+1 .V (a)τm−1,n + a τm−1,n+1 .V (a)τm,n = 0, (2.85) bV (a)τm,n .V (b)τm,n+1 − aV (b)τm,n .V (a)τm,n+1 (2.86) + (a − b)τm,n+1 .V (a)V (b)τm,n = 0. Proof Put (k, l, m) → (m + 1, l, −n + 1) and λ → ((m − 1)n−i−1 , m i ) in (2.79), we obtain the coefficients a −i of (2.84). Putting l → −n, λ → (m n ) in (2.80) we  have (2.85). Putting λ → (m n ) in (2.81) we have (2.86).

24

2 Padé Approximation for PVI

Date’s direct method Here, we explain Date’s direct method in soliton theory [5], which is based on a similar strategy to our method. Though Date’s original work deals with certain nonlinear differential equations, we will discuss in a discrete setting to show the algebraic structure more transparently. Let n (z) be a monic polynomial of degree n subjected to the linear relations n ( pi ) = ci n (qi ) (i = 1, . . . , n).

(2.87)

For generic parameters pi , qi , ci , the solution is unique. Let Vz be a shift operator acting on variables ci as z − qi ci . (2.88) Vz : ci → z − pi Consider a polynomial in z of degree at most n defined by ˜ n (z) := (z − a)Va n (z) − (z − b)Vb n (z) + (a − b) 

Vb (n (a)) n (z). (2.89) n (a)

˜ n (z) satisfies the same condition (2.87) as n (z), and hence It is easy to show that  ˜ ˜ n (z) vanishes at z = a, the conn (z) = Cn (z) with C independent of z. Since  ˜ stant C is zero, namely n (z) = 0 identically. ˜ Since (b) = 0 also, we have Vb (n (a)) Va (n (b)) = . n (a) n (b)

(2.90)

From this, we see that there exists a polynomial τn in {ci } (independent of z) such that n (z − pi )Vz (τn ) . (2.91) n (z) = i=1 τn Then, from (2.89) and (2.91), we have the Hirota–Miwa equation [15] (b − c)Va (τn )Vb Vc (τn ) + (abc cyclic) = 0.

(2.92)

Explicitly, the τ -functions τn can be written as τ0 τ1 τ2 τ3

= 1, = 1 + η1 , = 1 + η1 + η2 + a1,2 η1 η2 , 3  = 1 + i=1 ηi + i< j ai, j ηi η j + a1,2 a1,3 a2,3 η1 η2 η3 ,

and determined recursively by

(2.93)

2.5 More on the Schur Functions

25

 τn = τn−1 + ηn V p−1 Vqn (τn−1 ) = τn−1 + ηn (τn−1 ηi →ai,n ηi ), n where ηi = −ci

n  j (=i)=1

pi − p j ( pi − p j )(q j − qi ) . , ai, j = qi − p j ( pi − q j )( p j − qi )

(2.94)

(2.95)

These are well-known formulas of the soliton solutions for various soliton equations. This example shows that a suitable linear problem may give interesting solutions for nonlinear equations. Such a simple method to derive soliton solutions is known as Date’s direct method, which can be viewed as a shortcut of the inverse scattering method, or the degenerate limit of Krichever’s construction of the theta function solutions from the Baker–Akhiezer function. Similarly, in our approach, the Padé approximations give suitable linear problems to study the isomonodromic equations.

2.6

Appendix

In this appendix, we recall some basic facts on Painlevé and Garnier equations. PVI equation Painlevé equations are the very special nonlinear differential equations discovered by Painlevé and Gambier around 1900. Classically, they are classified into six equations PJ (J = I, . . . , VI). The most generic case among them is the sixth Painlevé equation PVI and others can be considered as its degenerations. For the general backgrounds of these equations we refer the reader to the textbook [17] and references therein. Here, we will give a very brief introduction to the equation PVI . The original form of PVI takes the following form: 11 1 1 2  1 1 1  q − q + + + + 2 q q −1 q −t t t −1 q −t β q(q − 1)(q − t)  t −1 t (t − 1) α+ 2 +γ . + +δ 2 2 2 t (t − 1) q (q − 1) (q − t)2

q  =

(2.96)

Here q = q(t) is the unknown function,  = dtd , and α, β, γ , δ are constant parameters. This equation can be written in a Hamiltonian form q = with the Hamiltonian

∂H , ∂p

p = −

∂H , ∂q

(2.97)

26

2 Padé Approximation for PVI

H=

 c d k q(q − 1)(q − t)  2 b p +( + + )p + , t (t − 1) q q −1 q −t q(q − 1)

(2.98)

−4k where α = (b+c+d) , β = − b2 , γ = c2 , δ = − d(d−2) . The Hamiltonian form is 2 2 fundamental for various studies of the PVI equation. Though H ( p, q) (2.98) looks like a rational function of ( p, q), it is actually a polynomial. Eliminating the variable p from Eq. (2.97), one can recover (2.96). To carry out this easily, we recommend the following procedure. First, consider the general case where H = H ( p, q, t) = A(q, t) p 2 + B(q, t) p + C(q, t) to obtain 2

q  =

2

2

 ∂ B  ∂  B2 1 ∂ A 2 1 ∂A  + q + q +A − 2C . 2 A ∂q A ∂t ∂t A ∂q 2 A

Then, specializing A, B, C as in (2.98), derive the expansion of

B2 2A

c2 c3 A B 2 c1 ( ) − 2C = + + + c4 q + c0 , 2 A q q −1 q −t

(2.99)

− 2C as (2.100)

−4k b where c1 = 2(t−1) , c2 = − c2t , c3 = d2 , c4 = (b+c+d) . The key to this computation 2t (t−1) is that we do not need to know the coefficient c0 , which is a little complicated. One of the most important aspects of the PVI is its characterization as the isomonodromic deformation due to R. Fuchs [7]. Namely, PVI describes the isomonodromic deformation of the Fuchsian equation on P1 with four regular singular points at {0, 1, t, ∞}. 2

2

2

2

Lax pair In previous sections, from the Padé problem for ψ(x) in (2.2), we derived two differential equations (2.3) and (2.22) of the form Ly = 0 :

L = ∂x2 + a(x)∂x + b(x),

By = 0 :

B = ∂t + G(x)∂x + F(x).

(2.101)

In relation to the Padé problem, the Eq. (2.101) can be considered in the following two different contexts: • (Situation on-Padé) m, n ∈ Z≥0 and the parameters q, p are some fixed functions of α, β, m, n and t, specified by the Padé problem. In this situation, the equations are automatically compatible by construction.

2.6 Appendix

27

• (Situation off-Padé) m, n ∈ C and the parameters q, p are variables or unknown functions to be solved from the compatibility condition. One can obtain various results by switching these two situations suitably. In this section, we study the Eq. (2.101) in an off-Padé situation. Then we have the following: Theorem 2.4 The differential equations (2.101) (i.e. (2.3) and (2.22)) are compatible if and only if the parameters q, p satisfy the following equation: qt =

∂H , ∂p

pt = −

∂H , ∂q

(2.102)

where H = − Res b(x) is given in (2.20). x=t This equation is the Hamiltonian form of PVI . Proof For simplicity, by a gauge transformation y(x) → w(x)y(x), w (x) = 1 a(x)w(x), we consider the case where 2 L y = {∂x2 + u(x)}y = 0, 1 1 u(x) = − a(x)2 − a  (x) + b(x). 4 2 We will show that qt =

∂K ∂K , μt = − , ∂μ ∂q

(2.103)

(2.104)

where α−1 β −1 1 m+n μ := Res u(x) = p − ( + + ), x=q 2 q q −1 q −t α−1 1 β −1 m+n K := − Res u(x) = H + ( + + ). x=t 2 t t −1 t −q

(2.105) (2.106)

Since the transformation ( p, q, H, t) → (μ, q, K , t) is symplectic, i.e. dp ∧ dq − d H ∧ dt = dμ ∧ dq − d K ∧ dt,

(2.107)

the Eqs. (2.102) and (2.104) are equivalent. A tedious but straightforward computation shows that for L of (2.103) and B of (2.101) the commutator [L , B] = L B − B L is given by [L , B] = (G  + 2F  )∂x + (F  − 2uG  − u t − Gu  ). Thus, the compatibility is given by F  = − 21 G  and

(2.108)

28

2 Padé Approximation for PVI

1 u t + ( ∂x3 + 2u∂x + u  )G = 0. 2

(2.109)

This gives a system of evolution equations for coefficients of u(x) with respect to the time variable t. To analyze the condition (2.109) further, we put the expansions of u(x), G(x) around z = x − q = 0, as u(x) =

∞ 

u n z n , G(x) =

n=−2

∞ 

ξn z n .

(2.110)

n=−1

Note that u −2 = − 43 (due to the exponents at x = q are 21 , − 32 ), and u −1 = μ, u 0 = −μ2 ,

(2.111)

due to the non-logarithmic condition (2.19). Hence we have the expansions 3 u t = − qt z −3 + μqt z −2 + μt z −1 + · · · , 2 ∞ ∞   1 1 ( ∂x3 + 2u∂x + u  )G = G  + (m + 2n)u m ξn z m+n−1 2 2

(2.112)

m=−2 n=−1

= (3 + 4u −2 )ξ−1 z −4 + (3u −1 ξ−1 + 2u −2 ξ0 )z −3 + (2u 0 ξ−1 + u −1 ξ0 )z −2 + (u 1 ξ−1 − u −1 ξ1 − 2u −2 ξ2 )z −1 + · · · .

Plugging these into (2.109), we obtain 

 ∂u G , x=q ∂μ   ∂u 3 G . μt = ξ2 − ξ1 μ + ξ−1 u 1 = − Res x=q 2 ∂q

qt = −2ξ−1 μ + ξ0 = Res

(2.113)

x(x − 1)(t − q) ∂u , the poles of G are only at t (t − 1)(x − q) ∂μ x = q, t. Then, by the residue theorem, we have In the case of (2.22), i.e. G(x) =

   ∂K ∂u ∂u G = − Res G = , qt = Res x=q x=t ∂μ ∂μ ∂μ     ∂u ∂u ∂K G = Res G =− . μt = − Res x=q x=t ∂q ∂q ∂q 

(2.114)

Thus we obtained (2.104), hence (2.102). Though these equations were derived as necessary conditions for the compatibility, one can show that they are sufficient by looking at the local computation at x = 0, 1, t, ∞ also. Namely, if the Eq. (2.104) is satisfied, then we can check that A := (LHS of (2.109)) has no poles on P1 and vanishes at x = ∞, hence A ≡ 0. 

2.6 Appendix

29

As we have seen above, the linear equations (2.101) are compatible if and only if the PVI Eq. (2.102) is satisfied. In this sense, they are called the Lax pair for PVI . Such a formulation of nonlinear equations gives a firm base for the inverse scattering method [1]. The equation L y = 0 in (2.101) is a Fuchsian equation on P1 \ {0, 1, t, ∞}, and the equation By = 0 describes its deformation. Note that yt has the same monodromy with y and yx since the coefficients of B are rational functions of x, hence the deformation is an isomonodromic deformation. As a result, the sixth Painlevé equation PVI was derived as the isomonodromic deformation of the equation (2.3). Such an isomonodromic interpretation of the Painlevé equations was first obtained by R. Fuchs [7]. From this point of view, we could foresee the relation between the Padé problem and the Painlevé equation from the beginning, since the monodromy of the functions P(x), ψ(x)Q(x) obviously does not depend on t.

Garnier system As a generalization of (2.103), consider the Fuchsian differential equation L y = {∂x2 + u(x)}y = 0, n  n+3  − 34 ci μi    Ki  + . u(x) = + − (x − λi )2 x − λi (x − ti )2 x − ti i=1 i=1

(2.115)

We set the regularity at x = ∞ and non-logarithmic conditions at x = λi : u(x) = O(x −4 ) (x → ∞), − 43 μi u(x) = + − μi2 + O((x − λi )1 ) (x → λi ), 2 (x − λi ) x − λi

(2.116)

namely n 

 3 λik {λi μi − (k + 1)} + tik {−ti K i + ci (k + 1)} = 0 (k = 0, ±1), 4 i=1 i=1

μ2j +

n+3

n   i(= j)=1

+

n+3   i=1

− 43 μi  + (λ j − λi )2 λ j − λi

ci Ki  = 0 ( j = 1, . . . , n). − (λ j − ti )2 λ j − ti

(2.117)

Solving these relations, one can determine K i as rational functions of {λi , μi , ti , ci }. The Garnier system is defined as the isomonodromic deformation of the equation (2.115). It is given as a rather complicated coupled partial differential equation for 2n unknown variables {λi , μi } w.r.t. (n + 3)-independent variables {ti }, The system is obtained as the compatibility of (2.115) and its deformation equation

30

2 Padé Approximation for PVI

{∂ti + G i (x)∂x + Fi (x)}y = 0, where

T (x) 1 (ti ) , Fi (x) = − G i (x), G i (x) =  2 T (ti ) (x − ti )(x) n n+3   (x) = (x − λi ), T (x) = (x − ti ). i=1

(2.118)

(2.119)

i=1

Fundamentally, the Garnier system can be described as a multi-time Hamiltonian system (2.59) where the Hamiltonian K i is the function determined by (2.117). We do not go into the proof of this here, but we will give a few remarks. (i) The Hamiltonian equation can be obtained through the same residue argument as the PVI case, where we do not need the explicit expression of K i . (ii) Due to the sl2 symmetry of P1 , three of the (n + 3) variables {ti } can be fixed to any values (e.g. 0, 1, ∞) and the system has effectively n-independent variables. The standard Garnier system is written in this way (see for example Appendix C of [14]). One recovers the Painlevé VI as the case n = 1. (iii) Though the Hamiltonian K i is rational function in {λi , μi }, it can be transformed into a polynomial Hamiltonian system by certain coordinate transformations [26]. Such a polynomial Hamiltonian system is obtained as the 2 × 2 case of the matrix Lax form (the Schlesinger system). We will study it in the last chapter (see Sect. 6.2).

Chapter 3

Padé Approximation for q-Painlevé/Garnier Equations

Abstract In this chapter, we consider the Padé approximation where the function ψ(x) is given by certain infinite products and apply it to the q-difference analog of the Painlevé/Garnier equations. As we will see later, the discrete analogs of the Padé approximation (Padé interpolations) are a more natural setting in which to approach the discrete Painlevé/Garnier equations. Hence the results in this chapter may be considered as a separate case; however, they show that there is a profound interplay between discrete and continuous cases.

3.1 Lax Pair for the q-Garnier Equation There are various discrete analogs of the Painlevé equations [50] and their geometric classification is given in [51]. The main target of this chapter is the q-PVI equation (the q-analog of the Painlevé VI) [19], see (3.29) below. Since the treatment is essentially the same also for its multivariable extensions (q-Garnier system [52]), we will start with the general setting corresponding to the q-Garnier system with 2N variables. The q-PVI equation corresponds to the N = 1 case. We assume that q ∈ C, 0 < |q| < 1, and we use the notation of a q-factorial symbol (x)s = (x; q)s , namely (x)∞ =

∞  (x)∞ (1 − q i x), (x)s = , (xq s )∞ i=0

(3.1)

and (x1 , x2 , . . . , xk )s = (x1 )s (x2 )s . . . (xk )s .

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd 2021 H. Nagao and Y. Yamada, Padé Methods for Painlevé Equations, SpringerBriefs in Mathematical Physics, https://doi.org/10.1007/978-981-16-2998-3_3

31

32

3 Padé Approximation for q-Painlevé/Garnier Equations

For the given function ψ(x) =

N +1  i=1

(ai x)∞ , (bi x)∞

(3.2)

we consider the Padé approximation ψ(x) =

P(x) + O(x m+n+1 ), Q(x)

(3.3)

where P(x), Q(x) are polynomials in x of degree m, n and ai , bi are complex parameters.

Using the relation log(x)∞ =

∞ 

log(1 − xq n ) = −

n=0

∞  ∞  (xq n )k n=0 k=1

k

=−

∞  k=1

xk , k(1 − q k )

(3.4)

the function ψ(x) can be written as ψ(x) = exp

N +1 ∞   bsk − ask k  x . k(1 − q k ) k=1 s=1

(3.5)

q-binomial theorem The following is well known and fundamental in q-analysis. Lemma 3.1 We have ∞  (ax)∞ (a)n n = x . (x)∞ (q) n n=0 1 Proof Due to the relation (qz)∞ = (1−z) (z)∞ , the LHS y(x) = q-difference equation (1 − ax)y(q x) = (1 − x)y(x).

(3.6) (ax)∞ (x)∞

satisfies the (3.7)

Solve this as a power series in x, with the initial condition y(0) = 1. Then we obtain the RHS.  The formula (3.6) is called the q-binomial theorem, because we have (q α x)∞ → (1 − x)−α (q → 1). (x)∞

(3.8)

3.1 Lax Pair for the q-Garnier Equation

33

 N +1 Putting ai = bi q −αi and taking a limit q → 1, we have ψ(x) = i=1 (1 − bi x)αi , which has the form ψ(x) (2.54) for the differential Garnier systems. So one can expect the Padé problem for ψ(x) (3.2) to be related to a certain q-analog of the Garnier system. Let P(x) and Q(x) be the polynomials of degree m and n determined by the Padé condition (1.10) normalized as P(0) = Q(0) = 1. For the parameter shift, we take T = Ta1 Tb1 such that   T( f ) = f  . (3.9) a1 →qa1 ,b1 →qb1

We also use the following up/down shift notation: f = T ( f ),

f = T −1 ( f ).

(3.10)

Let us consider the equations for y(x) satisfied by y(x) = P(x) and y(x) = ψ(x)Q(x). The main equation is the three-term relation L 1 between y(q x), y(x), y( qx ); however, it is a little difficult to determine the explicit form. So we will begin with the contiguity type relations, from which the L 1 equation will be considered in the next section. Theorem 3.1 The linear difference relation L 2 between y(x), y(qz), y(x), and the relation L 3 between y(x), y(x), y(x/q) satisfied by the functions y(x) = P(x) and y(x) = ψ(x)Q(x) are given as follows:1 L 2 : (g0 )1 F(x)y(x) − A1 (x)y(q x) + (b1 x)1 G(x)y(x) = 0, L3 : (

(3.11)

1 g0 x x x x )1 F( )y(x) + (a1 x)1 G( )y(x) − B1 ( )y( ) = 0, c q c q q q

(3.12)

where c = q m+n+1 . A(x), B(x), Ai (x), Bi (x), F(x), G(x) are the following polynomials in x: A(x) =

N +1  i=1

Ai (x) =

(ai x)1 , B(x) =

N +1 

(bi x)1 , F(x) = 1 +

N 

i=1

N −1  A(x) B(x) , Bi (x) = , G(x) = gi x i , (ai x)1 (bi x)1 i=0

fi x i ,

(3.13)

i=1

(3.14)

and f 1 , . . . , f N , g0 , . . . , g N −1 are some constants which play the role of dependent variables of the q-Garnier system. Proof The derivation of the equations (3.11), (3.12) is similar to the previous examples and we will show the outline of the computation. In this proof we use another up/down-shift notation such as We use the notation (z)1 = (1 − z). Though this is a non-standard use of the q-factorial symbol (3.1), it is sometimes useful to shorten the expressions.

1

34

3 Padé Approximation for q-Painlevé/Garnier Equations

  fx = f

x→q x

  fx = f 

,

x→x/q

.

(3.15)

By the definition, L 2 is given by    y yx y    = D1 y − D2 y x + D3 y = 0, L2 :  u ux u 

(3.16)

where D1 , D2 , D3 are Casorati determinants:  D1 = |u, ux |, Noting that

ψx ψ

=

B A

D2 = |u, u|,

D3 = |ux , u|, u =

P . ψQ

(3.17)

we have   P  D1 =   ψQ

Px ψ x Qx

   P    =ψ  Q 

Px B x Q A

    

(3.18)

ψ {B P Q x − A P x Q} A ψ x m+n+1 ψ x m+n+1 [N ] x = w0 F(x), = A A =

where x [N ] represents a polynomial in x of degree N , and w0 is a constant. Similarly 1 x)1 from ψψ = (b , we have (a1 x)1   P  D2 =   ψQ

P ψQ

   P    =ψ  Q 

    (b1 x)1 Q  (a1 x)1 P

(3.19)

ψ {(b1 x)1 P Q − (a1 x)1 P Q} (a1 x)1 ψ x m+n+1 [0] ψ A1 x m+n+1 w1 . = x = (a1 x)1 A

=

We also have   Px  D3 =   (ψ Q)x

P ψQ

      =ψ  

Px B x Q A

    (b1 x)1 Q  (a1 x)1 P

ψ(b1 x)1 {A1 P x Q − B1 P Q x } A ψ(b1 x)1 x m+n+1 ψ(b1 x)1 x m+n+1 [N −1] x w1 G(x), = = A A

=

with some constant w1 . Then the L 2 equation has the form

(3.20)

3.1 Lax Pair for the q-Garnier Equation

35

w0 F(x)y(x) − A1 (x)y(q x) + (b1 x)1 G(x)y(x) = 0. w1

(3.21)

0 We see that w = (g0 )1 from the solution y(x) = P(x) with P(0) = 1, hence we w1 obtain (3.11). For L 3 equation, its x-up shift L 3x is given by

 x x  y y y L 3x :  x x  = D1 y x + D3 y x − D2x y = 0, u u u

(3.22)

where the coefficients are 

(b1 x)1 ψ x m+n+1 ψ x m+n+1 F = D 1 = w0 w0 F, A (a1 x)1 (a1 q x)1 A1 ψ x m+n+1 D 3 = w1 (b1 x)1 G, A x

ψ x m+n+1 B ψ(q x)m+n+1 x D 2 = w1 = w1 . (a1 x)1 A (a1 q x)1

(3.23)

Hence L 3x takes the form w0 F y(q x) − (a1 q x)1 G(x)y(q x) + B1 (x)q m+n+1 y(x) = 0, w1 and, again from the solution y(x) = P(x) = 1 + O(x), we have and the expression (3.12) for L 3 .

w0 w1

(3.24)

= g0 − q m+n+1 

In the discrete case also, we will use the on/off-Padé situations (analogous to continuous case, §2.6) according to our purpose. And now, we switch to the off-Padé situation. Then we have the following: Proposition 3.1 In the off-Padé situation, the compatibility of the equations L 2 , L 3 in (3.11), (3.12) gives the following relations (c = q m+n+1 ) : 1 A1 (x)B1 (x) − (a1 x, b1 x)1 G(x)G(x) = 0 for F(x) = 0, c A1 (x)B1 (x) − (g0 ,

(g0 ,

g0 )1 F(x)F(x) = 0 for G(x) = 0, c

(3.25) (3.26)

N +1 N +1   g0 qa1 b1  qn  qm  )1 f N f N = g N −1 + (−bi ) g N −1 + (−ai ) . (3.27) c c a1 i=2 b1 i=2

Proof Equation (3.25) is obtained from L 2 (x) and L 3 (x), and (3.26) is from L 2 (x) and L 3 (q x). Considering the solution y(x) = P(x) = C x m + lower, the highest degree terms of L 2 (x), L 3 (q x) give

36

3 Padé Approximation for q-Painlevé/Garnier Equations N +1    (g0 )1 f N C = b1 g N −1 + q m (−ai ) C, i=2

(3.28)

N +1   g0 q a1 g N −1 + q n ( )1 f N C = (−bi ) C, c c i=2



hence we obtain (3.27).

One can prove [41] that the relations (3.25)−(3.27) are sufficient for the compatibility of L 2 , L 3 and these relations can be regarded as evolution equations of the q-Garnier system [52] along the direction T = Ta1 Tb1 . The equations are birational evolution equations for 2N variables (the coefficients of F(x) = 1 + f 1 x + · · · + f N x N and G(x) = g0 + · · · + g N −1 x N −1 ). Since their explicit forms are rather complicated in general, we will show simpler cases in the next example. Example For N = 1, in terms of the variables ( f, g) such that F(x) = 1 − f x and G(x) = g, we have the equations f f =

(g − q m ab21 )(g − q n ab21 )

qa1 b1 , (g − 1)(g − q m+n+1 ) ( f − a2 )( f − b2 ) m+n+1 q , gg = ( f − a1 )( f − b1 )

(3.29)

together with a1 = qa1 , b1 = qb1 . This system is known as the Jimbo–Sakai q-PVI [19] (the q-analog of PVI ), where m, n are not restricted to integers. For N = 2, the equation is a system with four variables in general. Under a suitable specialization, one can reduce it to a system of two variables [53] (see also [41]2 ). To do this, we put a constraint q m a1 a2 a3 = q n b1 b2 b3 on the parameters, then from (3.18) the degree of the polynomial F(x) is reduced from 2 to 1 (hence f 2 = 0). n m We can also put g1 = − q ab12 b3 = − q ba12 a3 consistently with (3.27). Then, in terms of variables f = − f 1 and g = − gg01 , the system can be written as ( f g − 1)( f g − 1) = ( f g − 1)( f g − 1) =

2

(1 −

f )(1 a2

−

(1 −

f )(1 a3 f )(1 a1

− −

f )(1 b2 f ) b1

−

f ) b3

,

(1 − a2 g)(1 − a3 g)(1 − b2 g)(1 − b3 g) , (1 − a4 g)(1 − b4 g)

In the second of equations (2.23) in [41], ( f g − 1) should read ( f g − 1).

(3.30)

3.1 Lax Pair for the q-Garnier Equation

37

Fig. 3.1 The singular point configuration for q-PVI (left) and q-P(E 6(1) ) (right)

where a4 =

q n b2 b3 , b4 a1

q −n−1 a2 a3 (a1 = qa1 , b1 = qb1 ). This system is known as the b1 (1) type E 6 . The Padé approximation problem related to this

=

q-Painlevé equation of equation was first studied in [16] (see also [37]). A convenient way to specify the (discrete) Painlevé equations is to consider the configuration of the singular points (i.e. the points where the birational mapping has indeterminacy). The configuration for (3.29) is on the four lines f = 0, f = ∞, g = 0, g = ∞, and that for (3.30) is on the two lines f = 0, g = 0 and a curve f g = 1 (see Fig. 3.1).

3.2 The L 1 Equation From (3.18) and its shift x → x/q, we see |u, ux | =

ψ(x) ψ(x) x m+n+1 x w0 x m+n+1 F(x), |ux , u| = F( ). x w0 ( ) A(x) B( q ) q q

(3.31)

Also it is not difficult to see |ux , ux | =

ψ(x) x m+n+1 x [2N +1] . A(x)B( qx )

(3.32)

Thus the L 1 equation takes the form x x x L 1 : A(x)F( )y(q x) − M(x)y(x) + q m+n+1 B( )F(x)y( ) = 0, q q q

(3.33)

38

3 Padé Approximation for q-Painlevé/Garnier Equations

where M(x) is a polynomial of degree 2N + 1. A characterization of M(x) is as follows: (i) Since the multiplicative exponents of the equation L 1 are 1, q m+n+1  N +1 bi m n 3 (at x = 0) and q , q i=1 ai (at x = ∞), the coefficients of the lowest and  N +1 the highest terms are C0 = 1 + q m+n+1 and C2N +1 = q −N f N q m i=1 (−ai ) +

 N +1 q n i=1 (−bi ) . (ii) For the roots λi of the polynomial F(x) and variables μi defined by y(qλi ) μi = (i = 1, . . . , N ), (3.34) y(λi ) we have the relations μi =

M(λi ) A(λi )F( λqi

)

,

M(qλi ) 1 . = q −m−n−1 μi B(λi )F(qλi )

(3.35)

The relations ensure the consistency of the L 1 equation at x = λi and x = qλi (a certain analog of the non-logarithmic condition; see the explanation after Remark 3.1). An explicit expression for the coefficient M(x) is as follows. From L 2 (3.11) we have 1 {A1 (x)y(q x) − (b1 x)1 G(x)y(x)}. y(x) = (3.36) (g0 )1 F(x) Using this, eliminate y(x) and y( qx ) from L 3 (3.12); we arrive at the L 1 equation with M(x) = c

(g0 ,

g0 ) F(x)F( qx )F(x) − c 1 G( qx )

A1 ( qx )B1 ( qx )F(x)

(3.37)

x x − (a1 x, b1 x)1 F( )G( ). q q Note that this M(x) is a polynomial in x due to the relation (3.26). Remark 3.1 The variables { f i , gi } and {λi , μi } are related by F(λi ) = 0 and A1 (λi )μi = (b1 λi )1 G(λi ) (i = 1, . . . , N ). The variables {λi , μi } have simple interpretation as the canonical coordinates defined through the Sklyanin’s magic recipe (see [54, 55] for example). Contrary to the variables { f i , gi }, the evolution equation in terms of {λi , μi } are not birational since we need roots λi of the algebraic equation F(x) = 0. q-analog of the non-logarithmic condition The constraint q m a1 a2 a3 = q n b1 b2 b3 for the reduction in the previous example can be interpreted as a degeneration condition of the exponents at x = ∞.

3

3.2 The L 1 Equation

39

Consider a q-difference equation of order n for unknown function y(x) a0 (x)y(x) + a1 (x)y(q x) + · · · + an (x)y(q n x) = 0,

(3.38)

where the coefficients ai (x) are assumed to be polynomials for simplicity. Since x = 0, ∞ are fixed points of the q-shift operation x → q x, the study of the local solutions of (3.38) has a different nature for x = 0, ∞ and other cases. Solutions around x = 0, ∞ We will concentrate on the case x = 0, since the case x = ∞ can be treated similarly using the local coordinate x −1 . To find the power series solutions around x = 0, we put yi (x) = x ρi

∞ 

ci,k x k (ci,0 = 1).

(3.39)

k=0

Then the exponents ρi are determined by the equation a0 (0) + a1 (0)t + · · · + an (0)t n = 0, t = q ρi .

(3.40)

If the exponents ρi are generic, the series solutions yi are unique and form a fundamental solution. Similar to the differential case, when some of the exponents have integer differences, there are two cases: (i) a special case where we still have power series solutions, (ii) a generic case where one should add logarithmic terms to some solutions. For example, consider the following equation with exponents ρi = 0, 2:   (q 2 + a1 x + a2 x 2 + · · · )y(x) + − (1 + q 2 ) + b1 x + b2 x 2 + · · · y(q x) + (1 + c1 x + c2 x 2 + · · · )y(q 2 x) = 0.

(3.41)

We look for a solution corresponding to ρ1 = 0 of the form y1 = 1 + u 1 x + u 2 x 2 + · · · . Substituting this into (3.41), we have the equations for the coefficients u 1 , u 2 , . . . as q(1 − q)2 u 1 = a1 + b1 + c1 , (3.42) 0 · u 2 = a2 + b2 + c2 + (a1 + qb1 + q 2 c1 )u 1 , . . . . If the RHS of the 2nd equation is zero (case (i)), the coefficient u 2 is free and we have two series solutions corresponding to ρi = 0, 2, otherwise (case (ii)) the solution has a logarithmic term. Solutions around x = c(= 0, ∞) We rewrite (3.38) as y(q n x) = −

1 {a0 (x)y(x) + · · · + an−1 (x)yn−1 (q n−1 x)}. an (x)

(3.43)

40

3 Padé Approximation for q-Painlevé/Garnier Equations

Using this recursively, y(q k x) (k ≥ n) can be expressed in terms of y(x), y(q x), · · · , y(q n−1 x). Since the denominator has factors an (x)an (q x) · · · an (q k−n x), the solution will have poles c, qc, q 2 c, · · · for the zeros c of the coefficient an (x). Such an infinite sequence of poles can be considered as a singularity (discrete analog of the branch cut). Similarly, the zeros of a0 (x) also could be a source of singularities (for y(q k x) (k