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Computational Techniques for the Summation of Series / Edition 1

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More About This Textbook

Overview

"This book collects in one volume the author’s considerable results in the area of the summation of series and their representation in closed form, and details the techniques by which they have been obtained... the calculations are given in plenty of detail, and closely related work which has appeared in a variety of places is conveniently collected together." —The Australian Mathematical Society Gazette

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Editorial Reviews

From the Publisher
From the reviews:

"This book collects in one volume the author’s considerable results in the area of the summation of series and their representation in closed form, and details the techniques by which they have been obtained. … the calculations are given in plenty of detail, and closely related work which has appeared in a variety of places is conveniently collected together. That the author passes on his extensive knowledge of the literature of results for series will also be valued by the interested scholar." (Katherine Seaton, The Australian Mathematical Society Gazette, 32:1, 2005)

"The author presents an unified treatment of summation of sums and series using function theoretic methods. … No book of this type exists which attempts to give a link, by developing a comprehensive method, between non-hypergeometric and hypergeometric summation. … This book is intended for use in the fields of applied mathematics, analysis, non-hypergeometric and hypergeometric summation, summation of series and automated techniques." (Antonio López-Carmona, Zentralblatt MATH, Vol. 1059 (10), 2005)

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Product Details

  • ISBN-13: 9780306478055
  • Publisher: Springer US
  • Publication date: 11/28/2003
  • Edition description: 2003
  • Edition number: 1
  • Pages: 189
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.50 (d)

Table of Contents

Preface
Acknowledgments
1 Some Methods for Closed Form Representation 1
1 Some Methods 1
2 A Tree Search Sum and Some Relations 14
2 Non-Hypergeometric Summation 31
1 Introduction 31
2 Method 32
3 Burmann's Theorem and Application 36
4 Differentiation and Integration 39
5 Forcing Terms 40
6 Multiple Delays, Mixed and Neutral Equations 42
7 Bruwier Series 43
8 Teletraffic Example 44
9 Neutron Behaviour Example 46
10 A Renewal Example 48
11 Ruin Problems in Compound Poisson Processes 50
12 A Grazing System 50
13 Zeros of the Transcendental Equation 51
14 Numerical Examples 53
15 Euler's Work 53
16 Jensen's Work 55
17 Ramanujan's Question 57
18 Cohen's Modifications and Extension 57
19 Conolly's Problem 60
3 Burmann's Theorem 63
1 Introduction 63
2 Burmann's Theorem and Proof 63
3 Convergence Region 69
4 Binomial Type Sums 73
1 Introduction 73
2 Problem Statement 73
3 A Recurrence Relation 74
4 Relations Between G[subscript k] (m) and F[subcript k+1] (m) 81
5 Generalization of the Euler Sum 87
1 Introduction 87
2 1-Dominant Zero 87
3 The k-Dominant Zeros Case 103
6 Hypergeometrics Summation: Fibonacci and Related Series 111
1 Introduction 111
2 The Difference-Delay System 111
3 The Infinite Sum 113
4 The Lagrange Form 114
5 Central Binomial Coefficients 116
6 Fibonacci, Related Polynomials and Products 123
7 Functional Forms 129
7 Sums and Products of Binomial Type 135
1 Introduction 135
2 Technique 136
3 Multiple Zeros 138
4 More Sums 142
5 Other Forcing Terms 144
8 Sums of Binomial Variation 147
1 Introduction 147
2 One Dominant Zero 147
3 Multiple Dominant Zeros 165
4 Zeros 169
5 Non-zero Forcing Terms 174
References 177
About the Author 185
Index 187
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