Table Of Content

This page intentionally left blank [50] Develop computer programs for simplifying sums that involve binomial coe–cients. 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•sek Herbert S. Wilf University of Ljubljana University of Pennsylvania Ljubljana, Slovenia Philadelphia, PA, USA Doron Zeilberger Temple University Philadelphia, PA, USA April 27, 1997 ii Contents Foreword vii A Quick Start ::: ix I Background 1 1 Proof Machines 3 1.1 Evolution of the province of human thought . . . . . . . . . . . . . . 3 1.2 Canonical and normal forms . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Polynomial identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Proofs by example? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Fibonacci identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.7 Symmetric function identities . . . . . . . . . . . . . . . . . . . . . . 12 1.8 Elliptic function identities . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Tightening the Target 17 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Human and computer proofs; an example . . . . . . . . . . . . . . . . 23 2.4 A Mathematica session . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5 A Maple session . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.6 Where we are and what happens next . . . . . . . . . . . . . . . . . . 30 2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 The Hypergeometric Database 33 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2 Hypergeometric series . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3 How to identify a series as hypergeometric . . . . . . . . . . . . . . . 35 3.4 Software that identifles hypergeometric series . . . . . . . . . . . . . . 39 iv CONTENTS 3.5 Some entries in the hypergeometric database . . . . . . . . . . . . . . 42 3.6 Using the database . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.7 Is there really a hypergeometric database? . . . . . . . . . . . . . . . 48 3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 II The Five Basic Algorithms 53 4 Sister Celine’s Method 55 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.2 Sister Mary Celine Fasenmyer . . . . . . . . . . . . . . . . . . . . . . 57 4.3 Sister Celine’s general algorithm . . . . . . . . . . . . . . . . . . . . . 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 Hypergeometrics to rationals to polynomials . . . . . . . . . . . . . . 75 5.3 The full algorithm: Step 2 . . . . . . . . . . . . . . . . . . . . . . . . 79 5.4 The full algorithm: Step 3 . . . . . . . . . . . . . . . . . . . . . . . . 84 5.5 More examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.6 Similarity among hypergeometric terms . . . . . . . . . . . . . . . . . 91 5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6 Zeilberger’s Algorithm 101 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.2 Existence of the telescoped recurrence . . . . . . . . . . . . . . . . . . 104 6.3 How the algorithm works . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.5 Use of the programs . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 7 The WZ Phenomenon 121 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 7.2 WZ proofs of the hypergeometric database . . . . . . . . . . . . . . . 126 7.3 Spinofis from the WZ method . . . . . . . . . . . . . . . . . . . . . . 127 7.4 Discovering new hypergeometric identities . . . . . . . . . . . . . . . 135 7.5 Software for the WZ method . . . . . . . . . . . . . . . . . . . . . . . 137 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 CONTENTS v 8 Algorithm Hyper 141 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.2 The ring of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 8.3 Polynomial solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 8.4 Hypergeometric solutions . . . . . . . . . . . . . . . . . . . . . . . . . 151 8.5 A Mathematica session . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.6 Finding all hypergeometric solutions . . . . . . . . . . . . . . . . . . 157 8.7 Finding all closed form solutions . . . . . . . . . . . . . . . . . . . . . 158 8.8 Some famous sequences that do not have closed form . . . . . . . . . 159 8.9 Inhomogeneous recurrences . . . . . . . . . . . . . . . . . . . . . . . . 161 8.10 Factorization of operators . . . . . . . . . . . . . . . . . . . . . . . . 162 8.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 III Epilogue 169 9 An Operator Algebra Viewpoint 171 9.1 Early history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.2 Linear difierence operators . . . . . . . . . . . . . . . . . . . . . . . . 172 9.3 Elimination in two variables . . . . . . . . . . . . . . . . . . . . . . . 177 9.4 Modifled elimination problem . . . . . . . . . . . . . . . . . . . . . . 180 9.5 Discrete holonomic functions . . . . . . . . . . . . . . . . . . . . . . . 184 9.6 Elimination in the ring of operators . . . . . . . . . . . . . . . . . . . 185 9.7 Beyond the holonomic paradigm . . . . . . . . . . . . . . . . . . . . . 185 9.8 Bi-basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.9 Creative anti-symmetrizing . . . . . . . . . . . . . . . . . . . . . . . . 188 9.10 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 9.11 Abel-type identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.12 Another semi-holonomic identity . . . . . . . . . . . . . . . . . . . . 193 9.13 The art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 A The WWW sites and the software 197 A.1 The Maple packages EKHAD and qEKHAD . . . . . . . . . . . . . . . . . 198 A.2 Mathematica programs . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Bibliography 201 Index 208 vi CONTENTS Foreword Science is what we understand well enough to explain to a computer. Art is everythingelsewedo. Duringthepastseveralyearsanimportantpartofmathematics hasbeentransformedfromanArttoaScience: Nolongerdoweneedto getabrilliant insight in order to evaluate sums of binomial coe–cients, 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 ofi two new problems that I was working on at the time I flrst 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 ‡ ·‡ · P methods just yesterday. Long ago I had run into the sum 2n¡2k 2k , which takes k n¡k k the values 1, 4, 16, 64 for n = 0, 1, 2, 3 so it must be 4n. Eventually I learned a tricky wayto provethatitis, indeed, 4n; butif Ihadknownthemethods inthisbook Icould have proved the identity immediately. Yesterday I was working on a harder problem ‡ · ‡ · whose answer was S = P 2n¡2k 2 2k 2. I didn’t recognize any pattern in the flrst n k n¡k k values 1, 8, 88, 1088, so I computed away with the Gosper-Zeilberger algorithm. In a few minutes I learned that n3S = 16(n¡ 1)(2n2¡2n+1)S ¡256(n¡1)3S . n 2 n¡1 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 stufi. ScienceadvanceswheneveranArtbecomesaScience. AndthestateoftheArtad- 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