Computational Techniques for the Summation of Series [2003 ed.] 0306478056, 9780306478055

Table of contents :
Contents
Preface
Acknowledgments
Chapter 1. SOME METHODS FOR CLOSED FORM REPRESENTATION
1. Some Methods
1.1 Introduction
1.2 Contour Integration
1.3 Use of Integral Equations
1.4 Wheelon's ResuIts
1.5 Hypergeometric Functions
2. A Tree Search Sum and Some Relations
2.1 Binomial Summation
2.2 Riordan
2.3 Method of Jonassen and Knuth
2.4 Method of Gessel
2.5 Method of Rousseau
2.6 Hypergeometrie Form
2.7 Snake Oil Method
2.8 Some Relations
2.9 Method of Sister Celine
2.10 Method of Creative Telescoping
2.11 WZ Pairs Method
Chapter 2. NON-HYPERGEOMETRIC SUMMATION
1. Introduction
2. Method
3. Bürmann's Theorem and AppIication
4. Differentiation and Integration
5. Forcing Terms
6. Multiple Delays, Mixed and Neutral Equations
7. Bruwier Series
8. Teletraffic Example
9. Neutron Behaviour Example
10. A Renewal Example
11. Ruin Problems in Compound Poisson Processes
12. A Grazing System
13. Zeros of the Transcendental Equation
14. Numerical Examples
15. Euler's Work
16. Jensen's Work
17. Ramanujan's Question
18. Cohen's Modification and Extension
19. Conolly's Problem
Chapter 3. BÜRMANN'S THEOREM
1. Introduction
2. Bürmann's Theorem and Proof
2.1 Applying Bürmann's Theorem
2.2 The Remainder
3. Convergence Region
3.1 Extension of the Series
Chapter 4. BINOMIAL TYPE SUMS
1. Introduction
2. Problem Statement
3. A Recurrence Relation
4. Relations Between G_k (m) and F_{k+1} (m)
Chapter 5. GENERALIZATION OF THE EULER SUM
1. Introduction
2. 1-Dominant Zero
2.1 The System
2.2 Q_{R,k} (0) Recurrences and Closed Forms
2.3 Lemma and Proof of Theorem 5.1
2.4 Extension of Results
2.5 Renewal Processes
3. The k-Dominant Zeros Case
3.1 The k-System
3.2 Examples
3.3 Extension
Chapter 6. HYPERGEOMETRIC SUMMATION: FIBONACCI AND RELATED SERIES
1. Introduction
2. The Difference-Delay System
3. The Infinite Sum
4. The Lagrange Form
5. Central Binomial Coefficients
5.1 Related Results
6. Fibonacci, Related Polynomials and Products
7. Functional Forms
Chapter 7. SUMS AND PRODUCTS OF BINOMIAL TYPE
1. Introduction
2. Technique
3. Multiple Zeros
4. More Sums
5. Other Forcing Terms
Chapter 8. SUMS OF BINOMIAL VARIATION
1. Introduction
2. One Dominant Zero
2.1 Recurrences
2.2 Proof of Conjecture
2.3 Hypergeometric Functions
2.4 Forcing Terms
2.5 Products of Central Binomial Coefficients
3. Multiple Dominant Zeros
3.1 The k Theorem
4. Zeros
4.1 Numerical Results and Special Cases
4.2 The Hypergeometric Connection
5. Non-zero Forcing Terms
References
1-13
14-30
31-49
50-67
68-87
88-106
107-115
About the Author
Index