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

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Wolfram Koepf

Hypergeometric Summation

An Algorithmic Approach to Summation and Special Function Identities

Wolfram Koepf

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English

Springer London Ltd
25 June 2014

Functional analysis & transforms; Differential calculus & equations; Combinatorics & graph theory; Mathematical & statistical software

Series: Universitext

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.

By: Wolfram Koepf
Imprint: Springer London Ltd
Country of Publication: United Kingdom
Edition: 2nd ed. 2014
Dimensions: Height: 235mm, Width: 155mm, Spine: 16mm
Weight: 4.569kg
ISBN: 9781447164630
ISBN 10: 1447164636
Series: Universitext
Pages: 279
Publication Date: 25 June 2014
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

Introduction.- The Gamma Function.- Hypergeometric Identities.- Hypergeometric Database.- Holonomic Recurrence Equations.- Gosper’s Algorithm.- The Wilf-Zeilberger Method.- Zeilberger’s Algorithm.- Extensions of the Algorithms.- Petkovˇsek’s and Van Hoeij’s Algorithm.- Differential Equations for Sums.- Hyperexponential Antiderivatives.- Holonomic Equations for Integrals.- Rodrigues Formulas and Generating Functions.

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.

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

“The book under review deals with the modern algorithmic techniques for hypergeometric summation, most of which were introduced in the 1990’s. … This well-written book should be recommended for anybody who is interested in binomial summations and special functions. It should also prove useful to researchers in mathematics and/or quantum physics working in topics which associate combinatorics of special (q-)functions … with current quantum mechanics issues.” (Christian Lavault, Mathematical Reviews, August, 2015)