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Special Values of the Hypergeometric Series

About this Title

Akihito Ebisu, Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo, 060-0810, Japan

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 248, Number 1177
ISBNs: 978-1-4704-2533-3 (print); 978-1-4704-4056-5 (online)
Published electronically: March 15, 2017
Keywords: Hypergeometric series, three term relation, special value, solving polynomial systems
MSC: Primary 33C05; Secondary 13P10, 13P15, 33C20, 33C65

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Table of Contents


  • 1. Introduction
  • 2. Preliminaries
  • 3. Derivation of special values
  • 4. Tables of special values
  • A. Some hypergeometric identities for generalized hypergeometric series and Appell-Lauricella hypergeometric series
  • Acknowledgments


In this paper, we present a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using this method, we get identities for the hypergeometric series $F(a,b;c;x)$; we show that values of $F(a,b;c;x)$ at some points $x$ can be expressed in terms of gamma functions, together with certain elementary functions. We tabulate the values of $F(a,b;c;x)$ that can be obtained with this method. We find that this set includes almost all previously known values and many previously unknown values.

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