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This book is of interest to mathematicians and computer scientists working in finite mathematics and combinatorics. It presents a breakthrough method for analyzing complex summations. Beautifully written, the book contains practical applications as well as conceptual developments that will have applications in other areas of mathematics. From the table of contents: \* Proof Machines \* Tightening the Target \* The Hypergeometric Database \* The Five Basic Algorithms: Sister Celine's Method, Gosper&'s Algorithm, Zeilberger's Algorithm, The WZ Phenomenon, Algorithm Hyper \* Epilogue: An Operator Algebra Viewpoint \* The WWW Sites and the Software (Maple and Mathematica) Each chapter contains an introduction to the subject and ends with a set of exercises. Cover Title Page Table of Contents Foreword A Quick Start I 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 Hypergeometric Database 3.1 Introduction 3.2 Hypergeometric series 3.3 How to identify a series as hypergeometric 3.4 Software that identifies hypergeometric series 3.5 Some entries in the hypergeometric database 3.6 Using the database 3.7 Is there really a hypergeometric database? 3.8 Exercises II The Five Basic Algorithms 4 Sister Celine’s Method 4.1 Introduction 4.2 Sister Mary Celine Fasenmyer 4.3 Sister Celine’s general algorithm 4.4 The Fundamental Theorem 4.5 Multivariate and “q” generalizations 4.6 Exercises 5 Gosper’s Algorithm 5.1 Introduction 5.2 Hypergeometrics to rationals to polynomials 5.3 The full algorithm: Step 2 5.4 The full algorithm: Step 3 5.5 More examples 5.6 Similarity among hypergeometric terms 5.7 Exercises 6 Zeilberger’s Algorithm 6.1 Introduction 6.2 Existence of the telescoped recurrence 6.3 How the algorithm works 6.4 Examples 6.5 Use of the programs 6.6 Exercises 7 The WZ Phenomenon 7.1 Introduction 7.2 WZ proofs of the hypergeometric database 7.3 Spinoffs from the WZ method 7.4 Discovering new hypergeometric identities 7.5 Software for the WZ method 7.6 Exercises 8 Algorithm Hyper 8.1 Introduction 8.2 The ring of sequences 8.3 Polynomial solutions 8.4 Hypergeometric solutions 8.5 A Mathematica session 8.6 Finding all hypergeometric solutions 8.7 Finding all closed form solutions 8.8 Some famous sequences that do not have closed form 8.9 Inhomogeneous recurrences 8.10 Factorization of operators 8.11 Exercises III Epilogue 9 An Operator Algebra Viewpoint 9.1 Early history 9.2 Linear difference operators 9.3 Elimination in two variables 9.4 Modified elimination problem 9.5 Discrete holonomic functions 9.6 Elimination in the ring of operators 9.7 Beyond the holonomic paradigm 9.8 Bi-basic equations 9.9 Creative anti-symmetrizing 9.10 Wavelets 9.11 Abel-type identities 9.12 Another semi-holonomic identity 9.13 The art 9.14 Exercises A The WWW sites and the software A.1 The Maple packages EKHAD and qEKHAD A.2 Mathematica programs Bibliography Index In the area of combinatorial identities computers are able to find very elegant proofs of difficult theorems unassisted by human intervention. This autonomy means that not only is it possible to use a PC to find such a proof, but also the proof can be checked easily. This book shows how several recently developed computer algorithms can master the difficult job of simplifying complex summations and if there is no such simplification they will prove this to be the case. The authors present the underlying mathematical theory of these methods, the principle theorems and proofs, and include a package of computer programs that can do these tasks This book is helpful for mathematicians and computer scientists working in finite mathematics. It presents a breakthrough method for analyzing complex summations, and contains practical applications and conceptual developments having applications in other areas of mathematics.