Computational Techniques for the Summation of Series

Transcription

1 Computational Techniques for the Summation of Series

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

3 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 ISBN (ebook) DOI / Series. 2. Summability theory. 3. Functional equations. 4. Functions of complex variables. I. Title. QA295.S '.243-dc ISBN Springer Science+Business Media New York Originally published by Kluwer Academic/Plenum Publishers, New York in 2003 Softcover reprint of the hardcover 1st edition 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.

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

5 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 VB

6 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 A Renewal Example Ruin Problems in Compound Poisson Processes A Grazing System Zeros of the Transcendental Equation Numerical Examples Euler's Work Jensen's Work Ramanujan's Question Cohen's Modification and Extension Conolly's Problem BÜRMANN'S THEOREM 63 1 Introduction 63 2 Bürmann's Theorem and Proof Applying Bürmann's Theorem The Remainder 68 3 Convergence Region Extension of the Series 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) GENERALIZATION OF THE EULER SUM 87 Introduction 87 2 I-Dominant Zero The System QR,k (0) Recurrences and Closed Forms Lemma and Proof of Theorem Extension of Results 99

7 Contents IX 2.5 Renewal Processes The k-dominant Zeros Case The k-system Examples Extension HYPERGEOMETRIC SUMMATION: FIBONACCI AND RELATED SERIES 111 Introduction The Difference-Delay System The Infinite Sum The Lagrange Form Central Binomial Coefficients Related Results Fibonacci, Related Polynomials and Products Functional Fom1s SUMS AND PRODUCTS OF BINOMIAL TYPE Introduction Technique Multiple Zeros More Sums Other Forcing Terms SUMS OF BINOMIAL VARIATION Introduction One Dominant Zero Recurrences Proof of Conjecture Hypergeometric Functions Forcing Terms Products of Central Binomial Coefficients Multiple Dominant Zeros The k Theorem Zeros Numerical Results and Special Cases The Hypergeometric Connection Non-zero Forcing Terms 174

8 x COMPUTATIONAL TECHNIQUES FOR THE SUMMATION OF SERIES References About the Author Index

9 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: 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

10 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.

11 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 A. SOFO

12 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