A=B potx

This page intentionally left blank [50] Develop computer programs for simplifying sums that involve binomial coefficients. Exercise 1.2.6.63 in The Art of Computer Programming, Volume 1: Fundamental Algorithms by Donald E. Knuth, Addison Wesley, Reading, Massachusetts, 1968. A=B Marko Petkov ˇ sek University of Ljubljana Ljubljana, Slovenia Herbert S. Wilf University of Pennsylvania Philadelphia, PA, USA Doron Zeilberger Temple University Philadelphia, PA, USA April 27, 1997 ii Contents Foreword vii A Quick Start ix IBackground 1 1 Proof Machines 3 1.1 Evolutionoftheprovinceofhumanthought 3 1.2 Canonicalandnormalforms 7 1.3 Polynomialidentities 8 1.4 Proofsbyexample? 9 1.5 Trigonometricidentities 11 1.6 Fibonacciidentities 12 1.7 Symmetricfunctionidentities 12 1.8 Ellipticfunctionidentities 13 2 Tightening the Target 17 2.1 Introduction 17 2.2 Identities 21 2.3 Humanandcomputerproofs;anexample 23 2.4 AMathematicasession 27 2.5 AMaplesession 29 2.6 Whereweareandwhathappensnext 30 2.7 Exercises 31 3 The Hypergeometric Database 33 3.1 Introduction 33 3.2 Hypergeometricseries 34 3.3 Howtoidentifyaseriesashypergeometric 35 3.4 Softwarethatidentifieshypergeometricseries 39 iv CONTENTS 3.5 Someentriesinthehypergeometricdatabase 42 3.6 Usingthedatabase 44 3.7 Istherereallyahypergeometricdatabase? 48 3.8 Exercises 50 II The Five Basic Algorithms 53 4 Sister Celine’s Method 55 4.1 Introduction 55 4.2 SisterMaryCelineFasenmyer 57 4.3 SisterCeline’sgeneralalgorithm 58 4.4 The Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . 64 4.5 Multivariate and “q”generalizations 70 4.6 Exercises 72 5 Gosper’s Algorithm 73 5.1 Introduction 73 5.2 Hypergeometricstorationalstopolynomials 75 5.3 Thefullalgorithm:Step2 79 5.4 Thefullalgorithm:Step3 84 5.5 Moreexamples 86 5.6 Similarityamonghypergeometricterms 91 5.7 Exercises 95 6 Zeilberger’s Algorithm 101 6.1 Introduction 101 6.2 Existenceofthetelescopedrecurrence 104 6.3 Howthealgorithmworks 106 6.4 Examples 109 6.5 Useoftheprograms 112 6.6 Exercises 118 7 The WZ Phenomenon 121 7.1 Introduction 121 7.2 WZproofsofthehypergeometricdatabase 126 7.3 SpinoffsfromtheWZmethod 127 7.4 Discoveringnewhypergeometricidentities 135 7.5 SoftwarefortheWZmethod 137 7.6 Exercises 140 CONTENTS v 8 Algorithm Hyper 141 8.1 Introduction 141 8.2 Theringofsequences 144 8.3 Polynomialsolutions 148 8.4 Hypergeometricsolutions 151 8.5 AMathematicasession 156 8.6 Findingallhypergeometricsolutions 157 8.7 Findingallclosedformsolutions 158 8.8 Some famous sequences that do not have closed form . . . . . . . . . 159 8.9 Inhomogeneousrecurrences 161 8.10Factorizationofoperators 162 8.11Exercises 164 III Epilogue 169 9 An Operator Algebra Viewpoint 171 9.1 Earlyhistory 171 9.2 Lineardifferenceoperators 172 9.3 Eliminationintwovariables 177 9.4 Modifiedeliminationproblem 180 9.5 Discreteholonomicfunctions 184 9.6 Eliminationintheringofoperators 185 9.7 Beyondtheholonomicparadigm 185 9.8 Bi-basicequations 187 9.9 Creativeanti-symmetrizing 188 9.10Wavelets 190 9.11Abel-typeidentities 191 9.12Anothersemi-holonomicidentity 193 9.13Theart 193 9.14Exercises 195 A The WWW sites and the software 197 A.1 The Maple packages EKHAD and qEKHAD 198 A.2 Mathematicaprograms 199 Bibliography 201 Index 208 vi CONTENTS Foreword Science is what we understand well enough to explain to a computer. Art is everything else we do. During the past several years an important part of mathematics has been transformed from an Art to a Science: No longer do we need to get a brilliant insight in order to evaluate sums of binomial coefficients, and many similar formulas that arise frequently in practice; we can now follow a mechanical procedure and discover the answers quite systematically. I fell in love with these procedures as soon as I learned them, because they worked for me immediately. Not only did they dispose of sums that I had wrestled with long and hard in the past, they also knocked off two new problems that I was working on at the time I first tried them. The success rate was astonishing. In fact, like a child with a new toy, I can’t resist mentioning how I used the new methods just yesterday. Long ago I had run into the sum  k  2n−2k n−k  2k k  ,whichtakes the values 1, 4, 16, 64 for n =0,1,2,3soitmustbe4 n . Eventually I learned a tricky way to prove that it is, indeed, 4 n ; but if I had known the methods in this book I could have proved the identity immediately. Yesterday I was working on a harder problem whose answer was S n =  k  2n−2k n−k  2  2k k  2 . I didn’t recognize any pattern in the first values 1, 8, 88, 1088, so I computed away with the Gosper-Zeilberger algorithm. In afewminutesIlearnedthatn 3 S n =16(n− 1 2 )(2n 2 −2n +1)S n−1 −256(n −1) 3 S n−2 . Notice that the algorithm doesn’t just verify a conjectured identity “A = B”. It also answers the question “What is A?”, when we haven’t been able to formulate a decent conjecture. The answer in the example just considered is a nonobvious recurrence from which it is possible to rule out any simple form for S n . I’m especially pleased to see the appearance of this book, because its authors have not only played key roles in the new developments, they are also master expositors of mathematics. It is always a treat to read their publications, especially when they are discussing really important stuff. Science advances whenever an Art becomes a Science. And the state of the Art ad- vances too, because people always leap into new territory once they have understood more about the old. This book will help you reach new frontiers. Donald E. Knuth Stanford University 20 May 1995 viii CONTENTS . Algorithms by Donald E. Knuth, Addison Wesley, Reading, Massachusetts, 1968. A=B Marko Petkov ˇ sek University of Ljubljana Ljubljana, Slovenia Herbert S.