Computational Techniques for the Summation of Series

Computational Techniques for the Summation of Series

Computational Techniques for the Summation of Series Anthony Sofa School of Computer Science and Mathematics Victoria University Melboume, Australia Springer-Science+Business Media, LLC

Library of Congress Cataloging-in-Publication Data Sofo, Anthony.- Computational techniques for the summation of series/anthony Sofo. p. cm. Includes bibliographical references and index. ISBN 978-1-4613-4904-4 ISBN 978-1-4615-0057-5 (ebook) DOI 10.1007/978-1-4615-0057-5 1. Series. 2. Summability theory. 3. Functional equations. 4. Functions of complex variables. I. Title. QA295.S64122003 515'.243-dc22 2003054692 ISBN 978-1-4613-4904-4 2003 Springer Science+Business Media New York Originally published by Kluwer Academic/Plenum Publishers, New York in 2003 Softcover reprint of the hardcover 1st edition 2003 10 9 8 7 6 5 4 3 2 1 A c.i.p. record for this book is available from the Library of Congress All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

For their time and understanding I dedicate this text to my supportive wife Lucia and our wonderful children Fiona, Jessica, Clara and Matthew.

Contents Preface Acknowledgments xi xv 1. SOME METHODS FOR CLOSED FORM REPRESENTATION Some Methods 1.1 Introduction 1.2 Contour Integration 1.3 Use of Integral Equations 1.4 Wheelon 's ResuIts 1.5 Hypergeometrie 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 2. NON-HYPERGEOMETRIC SUMMATION 1 Introduction 2 Method 3 Bürmann's Theorem and AppIication 4 Differentiation and Integration 3 4 7 12 14 14 15 16 17 17 18 21 23 25 26 26 31 31 32 36 39 VB

viii COMPUTATIONAL TECHNIQUES FOR THE SUMMATION OF SERIES 5 Forcing Terms 40 6 Multiple Dclays, 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 Modification and Extension 57 19 Conolly's Problem 60 3. BÜRMANN'S THEOREM 63 1 Introduction 63 2 Bürmann's Theorem and Proof 63 2.1 Applying Bürmann's Theorem 67 2.2 The Remainder 68 3 Convergence Region 69 3.1 Extension of the Series 70 4. BINOMIAL TYPE SUMS 73 1 Introduction 73 2 Problem Statement 73 3 A Recurrence Relation 74 4 Relations Between G k (m) and Fk+ 1 (m) 81 5. GENERALIZATION OF THE EULER SUM 87 Introduction 87 2 I-Dominant Zero 87 2.1 The System 87 2.2 QR,k (0) Recurrences and Closed Forms 91 2.3 Lemma and Proof of Theorem 5.1 96 2.4 Extension of Results 99

Contents IX 2.5 Renewal Processes 102 3 The k-dominant Zeros Case 103 3.1 The k-system 103 3.2 Examples 107 3.3 Extension 108 6. HYPERGEOMETRIC SUMMATION: FIBONACCI AND RELATED SERIES 111 Introduction 111 2 The Difference-Delay System 111 3 The Infinite Sum 113 4 The Lagrange Form 114 5 Central Binomial Coefficients 116 5.1 Related Results 120 6 Fibonacci, Related Polynomials and Products 123 7 Functional Fom1s 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 2.1 Recurrences 149 2.2 Proof of Conjecture 152 2.3 Hypergeometric Functions 157 2.4 Forcing Terms 160 2.5 Products of Central Binomial Coefficients 161 3 Multiple Dominant Zeros 165 3.1 The k Theorem 166 4 Zeros 169 4.1 Numerical Results and Special Cases 172 4.2 The Hypergeometric Connection 173 5 Non-zero Forcing Terms 174

x COMPUTATIONAL TECHNIQUES FOR THE SUMMATION OF SERIES References About the Author Index 177 185 187

Preface Over the last twenty years or so, work on the c10sed form representation of sums and series has prospered and flourished. Recently, two great books dealing with the summation of series have appeared on the market. The book "A = B" by Petkovsek, Wilf and Zeilberger (1996) expounds the theory of hypergeometric summation and has given a great impetus to research in this area. The more recent book, "Hypergeometric summation: an algorithmic approach to summation and special function identities" by Koepf (1998) gives up-to-date algorithmic techniques for summation and examples are worked out using Maple programs. Another large group working in the area of c10sed form representation for sums and series is the Centre for Experimental and Constructive Mathematics (CECM) in Canada, led by the brilliant Professor J. Borwein. Many good papers may be viewed on the CECM website at: http://www.cecm.sfu.ca/. The main aim of this book is to present a unified treatment of summation of sums and series using function theoretic methods. We develop a tecimique, based on residue theory, that is useful for the summation of series of both non-hypergeometric and hypergeometric type. This book is intended to complement the books of Koepf and Petkovsek, Wilf and Zeilberger, it gives an extra comprehensive perspective on the many methods and procedures that are available for the summation of series. To the author's knowledge, no book of this type exists which attempts to give a link, by developing a comprehensive method, between non-hypergeometric and hypergeometric summation. The book has intentionally not been written as an algorithmic approach to summation, no doubt this will be done by other authors. In particular the book develops computational techniques for the summation of series. To put the book into context, Chapter I is an introductory one in which some methods for closed form summation are given. Methods dealing with residue theory are discussed and various results are extended. The second part XI

XII COMPUTATIONAL TECHNIQUES FOR THE SUMMATION OF SERIES ofchapter 1 investigates a partieular tree search sum and in the process develops some new identities. Chapter 2 investigates non-hypergeometrie summation. We introduce our function theoretic methods by considering a first order differential-difference equation. By the use oflaplace transform theory, an Abel type series is generated and summed in closed form. The series, it will be shown, arises in a number of different areas including teletraffic problems, neutron behaviour, renewal processes, risk theory, grazing systems and demographie problems. Related works to this area of study will be considered, including Euler's and Jensen's investigation, Ramanujan's question, Cohen's modification and Conolly's problem. In Chapter 3, we give a detailed proofofbürmann's Theorem and apply it to the Abel type series generated in Chapter 2. In Chapter 4, we consider Binomial type sums with some parameters and prove that they may be represented in polynomial form. The binomial type sums will be applied in the generalisation of the results obtained in Chapter 2. Moreover, the binomial type sums have a connection with Stirling numbers of the second kind and may be applied to problems in multinomial distributions. The results of Chapter 2 are generalised in Chapter 5. The generalised series, it will be proved, may be expressed in closed form which depend on the dominant zero of an associated transcendental function. A connection with renewal processes will also be made. We then prove that more general Abel type series may be expressed in closed form which depend on a multiple number of dominant zeros of an associated transcendental function. In Chapter 6, a first order difference-delay system is considered and by the use of Z -transform theory generate an infinite series which it will be proved, by the use of residue theory, may be represented in closed for. The work of Jensen will be considered and some work on central binomial coefficients will be undertaken. A development of Fibonacci and related polynomials will also be given. In Chapter 7, we consider arbitrary order forced difference-delay systems from which finite binomial type sums are generated. By considering multiple zeros of an associated polynomial characteristic function, many binomial type sums are represented in closed form. In Chapter 8, we extend the results of Chapter 6. In particular, binomial type series with free parameters are expressed in closed form that depend on the dominant zero of an associated polynomial characteristic function. A connection between the binomial sums and generalised hypergeometrie functions is made and particular cases, including Kummer's identity, are extracted. We further generate binomial type sums and prove that they may be represented in closed form in terms of k-dominant zeros of an associated characteristic function. Some numerical results and special cases are discussed.

PREFACE X 111 The present book is intended for use in the fields of applied mathematics, analysis, non-hypergeometric and hypergeometric summation, summation of series and automated techniques. Melbourne March 2003. A. SOFO

Acknowledgments I would like to thank Associate Professor Pietro Cerone for the many hours of constructive discourse on this subject. My thanks also go to Professor George Anastassiou, from the University of Memphis, USA, and to my colleagues from the Research Group in Mathematical Inequalities and Applications at Victoria University, Professor S.S. Dragomir and Dr. 1. Roumeliotis. xv