Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities

✍ Scribed by Wolfram Koepf (auth.)

• Publisher: Springer-Verlag London
• Year: 2014
• Tongue: English
• Leaves: 290
• Series: Universitext
• Edition: 1
• Category: Library

✦ Synopsis

Modern algorithmic techniques for summation, most of which were introduced in the 1990s, are developed here and carefully implemented in the computer algebra system Maple™.

The algorithms of Fasenmyer, Gosper, Zeilberger, Petkovšek and van Hoeij for hypergeometric summation and recurrence equations, efficient multivariate summation as well as q-analogues of the above algorithms are covered. Similar algorithms concerning differential equations are considered. An equivalent theory of hyperexponential integration due to Almkvist and Zeilberger completes the book.

The combination of these results gives orthogonal polynomials and (hypergeometric and q-hypergeometric) special functions a solid algorithmic foundation. Hence, many examples from this very active field are given.

The materials covered are suitable for an introductory course on algorithmic summation and will appeal to students and researchers alike.

✦ Table of Contents

Front Matter....Pages i-xvii
The Gamma Function....Pages 1-10
Hypergeometric Identities....Pages 11-33
Hypergeometric Database....Pages 35-48
Holonomic Recurrence Equations....Pages 49-77
Gosper’s Algorithm....Pages 79-101
The Wilf-Zeilberger Method....Pages 103-116
Zeilberger’s Algorithm....Pages 117-151
Extensions of the Algorithms....Pages 153-168
Petkovšek’s and van Hoeij’s Algorithm....Pages 169-204
Differential Equations for Sums....Pages 205-225
Hyperexponential Antiderivatives....Pages 227-237
Holonomic Equations for Integrals....Pages 239-254
Rodrigues Formulas and Generating Functions....Pages 255-269
Back Matter....Pages 271-279

✦ Subjects

Algorithms; Mathematical Software; Special Functions; Ordinary Differential Equations; Combinatorics