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[50] puter programs for simplifying sums
that involve binomial coefficients.
Exercise in
The Art puter Programming, Volume 1: Fundamental Algorithms
by Donald E. Knuth,
Addison Wesley, Reading, Massachusetts, 1968.
A=B
Marko Petkovˇsek 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
Evolution of the province of human thought . . . . . . . . . . . . . . 3
Canonical and normal forms . . . . . . . . . . . . . . . . . . . . . . . 7
........................... 8
?............................ 9
Trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . 11
............................ 12
Symmetric function identities . . . . . . . . . . . . . . . . . . . . . . 12
Elliptic function identities . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Tightening the Target 17
................................ 17
.................................. 21
Human puter proofs; an example . . . . . . . . . . . . . . . . 24
.......................... 27
.............................. 29
Where we are and what happens next . . . . . . . . . . . . . . . . . . 30
.................................. 31
3 The Hypergeometric Database 33
................................ 33
........................... 34
How to identify a series as hypergeometric . . . . . . . . . . . . . . . 35
Software that identifies hypergeometric series . . . . . . . . . . . . . . 39
iv CONTENTS
Some entries in the hypergeometric database . . . . . . . . . . . . . . 42