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

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Springer, Jun 10, 2014 - Computers - 279 pages

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

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.

 

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Contents

1 The Gamma Function
1
2 Hypergeometric Identities
11
3 Hypergeometric Database
34
4 Holonomic Recurrence Equations
49
5 Gospers Algorithm
79
6 The WilfZeilberger Method
102
7 Zeilbergers Algorithm
117
8 Extensions of the Algorithms
153
9 Petkovšeks and van Hoeijs Algorithm
169
10 Differential Equations for Sums
205
11 Hyperexponential Antiderivatives
226
12 Holonomic Equations for Integrals
239
13 Rodrigues Formulas and Generating Functions
255
Index
270
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About the author (2014)

Prof. Dr. Wolfram Koepf is Professor for Computational Mathematics at the University of Kassel. He started his research in geometric function theory, switching towards orthogonal polynomials and special functions and towards computer algebra. In the 1990s he has written several books about the use of computer algebra in math education, followed by the first edition of his monograph Hypergeometric Summation. In 2006 his German language text book Computeralgebra appeared. Between 2002 and 2010 he was the Chairman of the Fachgruppe Computeralgebra , the largest computer algebra group world-wide, in 2010 he served as the General Chair of the most important international computer algebra symposium ISSAC in Munich. Since 2010 he serves as PC chair of the conference series CASC. As a member of the executive committee of the German Mathematical Union (DMV) he is the responsible editor of the web resource Mathematik.de.

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