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[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

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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 . . . . . . . . . . . . . . . . 24 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 identifies hypergeometric series . . . . . . . . . . . . . . 39

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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 Spinoffs 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 143 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.2 The ring of sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 8.3 Polynomial solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8.4 Hypergeometric solutions . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.5 A Mathematica session . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8.6 Finding all hypergeometric solutions . . . . . . . . . . . . . . . . . . 159 8.7 Finding all closed form solutions . . . . . . . . . . . . . . . . . . . . . 160 8.8 Some famous sequences that do not have closed form . . . . . . . . . 161 8.9 Inhomogeneous recurrences . . . . . . . . . . . . . . . . . . . . . . . . 163 8.10 Factorization of operators . . . . . . . . . . . . . . . . . . . . . . . . 164 8.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 III Epilogue 171 9 An Operator Algebra Viewpoint 173 9.1 Early history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.2 Linear difference operators . . . . . . . . . . . . . . . . . . . . . . . . 174 9.3 Elimination in two variables . . . . . . . . . . . . . . . . . . . . . . . 179 9.4 Modified elimination problem . . . . . . . . . . . . . . . . . . . . . . 182 9.5 Discrete holonomic functions . . . . . . . . . . . . . . . . . . . . . . . 186 9.6 Elimination in the ring of operators . . . . . . . . . . . . . . . . . . . 187 9.7 Beyond the holonomic paradigm . . . . . . . . . . . . . . . . . . . . . 187 9.8 Bi-basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 9.9 Creative anti-symmetrizing . . . . . . . . . . . . . . . . . . . . . . . . 190 9.10 Wavelets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 9.11 Abel-type identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 9.12 Another semi-holonomic identity . . . . . . . . . . . . . . . . . . . . 195 9.13 The art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 9.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 A The WWW sites and the software 199 A.1 The Maple packages EKHAD and qEKHAD . . . . . . . . . . . . . . . . . 200 A.2

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