Đang chuẩn bị liên kết để tải về tài liệu:
Develop computer programs for simplifying sums that involve binomial coefficients: The Art of Computer Programming, Volume 1: Fundamental Algorithms

Science is what we understand well enough to explain to a computer. Art is everything else we do. During the past several years an important part of mathematics has been transformed from an Art to a Science: No longer do we need to get a brilliant insight in order to evaluate sums of binomial coefficients, and many similar formulas that arise frequently in practice; we can now follow a mechanical procedure and discover the answers quite systematically. | This page intentionally left blank [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 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 1.1 Evolution of the province of human thought . . . . . . . . . . . . . . 3 1.2 Canonical and normal forms . . . . . . . . . . . . . . . . . . . . . . . 7 1.3Polynomialidentities. 8 1.4Proofsbyexample? 9 1.5 Trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6Fibonacciidentities 12 1.7 Symmetric function identities . . . . . . . . . . . . . . . . . . . . . . 12 1.8 Elliptic function identities . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Tightening the Target 17 2.1Introduction 17 2.2Identities 21 2.3 Human and computer proofs; an example . . . . . . . . . . . . . . . . 24 2.4AMathematicasession 27 2.5AMaplesession 29 2.6 Where we are and what happens next . . . . . . . . . . . . . . . . . . 30 2.7Exercises 31 3 The Hypergeometric Database 33 3.1Introduction 33 3.2Hypergeometricseries. 34 3.3 How to identify a series as hypergeometric . . . . . . . . . . . . . . . 35 3.4 Software that identifies hypergeometric series . . . . . . . . . . . . . . 39 iv CONTENTS 3.5 Some entries in the hypergeometric database . . . . . . . . . . . . . . 42 3.6Usingthedatabase 44 3.7 Is