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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.
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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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ii
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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 P o lyn om i a lid en t i t i e s........................... 8 1 . 4 P r oo f sb yex amp l e ?............................ 9 1.5 Trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . 11 1 . 6 F ibon a c c iid t i t i e s............................ 1 2 1.7 Symmetric function identities . . . . . . . . . . . . . . . . . . . . . . 12 1.8 Elliptic function identities . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Tightening the Target 17 2 . 1 In t r odu c t i on. ............................... 1 7 2 . 2 Id t i t i e s.................................. 2 1 2.3 Human and computer proofs; an example . . . . . . . . . . . . . . . . 24 2 . 4 AM a th em a t i c as e s s i ......................... 2 7 2 . 5 AM ap l es e s s i ............................. 2 9 2.6 Where we are and what happens next . . . . . . . . . . . . . . . . . . 30 2 . 7 Ex e r c i s e 3 1 3 The Hypergeometric Database 33 3 . t r c t i 3 3 3 . 2 Hyp e r g e e t r i cs e r i e 3 4 3.3 How to identify a series as hypergeometric . . . . . . . . . . . . . . . 35 3.4 Software that identi±es hypergeometric series . . . . . . . . . . . . . . 39
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iv CONTENTS 3.5 Some entries in the hypergeometric database . . . . . . . . . . . . . . 42 3 . 6 U s in gth ed a t ab a s e ............................ 4 4 3.7 Is there really a hypergeometric database? . . . . . . . . . . . . . . . 48 3 . 8 Ex e r c i s e s.................................. 5 0 II The Five Basic Algorithms 53 4 Sister Celine’s Method 55 4 . 1 In t r odu c t i on. ............................... 5 5 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 Ex e r c i s e 7 2 5 Gosper’s Algorithm 73 5 . t r c t i 7 3 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 M o r eex amp l e s .............................. 8 6 5.6 Similarity among hypergeometric terms . . . . . . . . . . . . . . . . . 91 5 . 7 Ex e r c i s e 9 5 6 Zeilberger’s Algorithm 101 6 . t r c t i 1 0 1 6.2 Existence of the telescoped recurrence . . . . . . . . . . . . . . . . . . 104 6 . 3 H owth ea l g o r i thmw o rk s......................... 1 0 6 6 . 4 Ex l e s ................................. 1 0 9 6 . 5 U s eo fth ep r o g r am s ........................... 1 1 2 6 . e r c i s e 1 1 8 7 The WZ Phenomenon 121 7 . t r c t i 1 2 1 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 . e r c i s e 1 4 0
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CONTENTS v 8 Algorithm Hyper 143 8 . 1 In t r odu c t i on. ............................... 1 4 3 8 . 2 Th er in go fs equ en c e s........................... 1 4 6 8 . 3 P o lyn om i a ls o lu t i on s ........................... 1 5 0 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 . 1 1Ex e r c i s e s..................................
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This note was uploaded on 05/05/2011 for the course ECON 101 taught by Professor Prof.sk during the Spring '09 term at Waseda University.

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