The Art of Computer Programming

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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 Petkovˇek s Herbert S. Wilf

University of Ljubljana
Ljubljana, Slovenia

University of Pennsylvania
Philadelphia, PA, USA

Doron Zeilberger
Temple University
Philadelphia, PA, USA

April 27, 1997

ii

Contents
Foreword

vii

A Quick Start . . .

ix

I

1

Background

1 Proof Machines
1.1 Evolution of the province of human thought
1.2 Canonical and normal forms . . . . . . . . .
1.3 Polynomial identities . . . . . . . . . . . . .
1.4 Proofs by example? . . . . . . . . . . . . . .
1.5 Trigonometric identities . . . . . . . . . . .
1.6 Fibonacci identities . . . . . . . . . . . . . .
1.7 Symmetric function identities . . . . . . . .
1.8 Elliptic function identities . . . . . . . . . .
2 Tightening the Target
2.1 Introduction . . . . . . . . . . . . . . . .
2.2 Identities . . . . . . . . . . . . . . . . . .
2.3 Human and computer proofs; an example
2.4 A Mathematica session . . . . . . . . . .
2.5 A Maple session . . . . . . . . . . . . . .
2.6 Where we are and what happens next . .
2.7 Exercises . . . . . . . . . . . . . . . . . .
3 The
3.1
3.2
3.3
3.4

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Hypergeometric Database
Introduction . . . . . . . . . . . . . . . . . . .
Hypergeometric series . . . . . . . . . . . . . .
How to identify a series as hypergeometric . .
Software that identifies hypergeometric series .

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Bibliography: Math. Phys. 11 (1971), 324–330. Transl. from Zh. vychisl. mat. mat. fiz. 29, 1611–1620. with polynomial coefficients, in: T. Levelt, ed., Proc. ISSAC ’95, ACM Press, New York, 1995, 285–289. linear operator equations, in: T. Levelt, ed., Proc. ISSAC ’95, ACM Press, New York, 1995, 290–296. and difference equations, in: J. von zur Gathen, ed., Proc. ISSAC ’94 , ACM Press, New York, 1994, 169–174. squares, Amer. Math. Monthly 100 (1993), 274–276. [Andr93] Andrews, George E., Pfaff’s method (I): The Mills-Robbins-Rumsey determinant, preprint, 1993. Winston, New York, 1961. [BeO78] Bender, E., and Orszag, S.A., Advanced Mathematical Methods for Scientists and Engineers, New York: McGraw-Hill, 1978. e LACIM, UQAM, Montr´al, 1991. Ian Stewart, Math. Intell. 15, #4 (Fall 1993), 71–73. [Buch76] Buchberger, Bruno, Theoretical basis for the reduction of polynomials to canonical form, SIGSAM Bulletin 39 (Aug. 1976), 19–24. r´arrangements, Lecture Notes Math. 85, Springer, Berlin, 1969. Ast´risque 206 (1992), 41–91, SMF. 17 (1970), 385–396. [Cipr89] Cipra, Barry, How the Grinch stole mathematics, Science 245 (August 11, 1989), 595. [Cohn65] Cohn, R. M., Difference Algebra, Interscience Publishers, New York, 1965. D. Reidel Publ. Co., Dordrecht-Holland, Transl. of Analyse Combinatoire, Tomes I et II , Presses Universitaires de France, Paris, 1974. [Daub92] Daubechies, Ingrid, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. [Dixo03] Dixon, A. C., Summation of a certain series, Proc. London Math. Soc. (1) 35 (1903), 285–289.