Special values of the hypergeometric series

TL;DR

This paper introduces a new method for deriving identities of hypergeometric series, enabling the expression of their special values in terms of gamma functions and elementary functions, covering known and new cases.

Contribution

The paper presents a novel approach to find identities for various hypergeometric series, expanding the set of known special values and unifying previous results.

Findings

Identifies new hypergeometric identities using the proposed method.

Expresses hypergeometric series values in terms of gamma functions and elementary functions.

Includes almost all previously known and many new special values.

Abstract

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. Key Words and Phrases: hypergeometric series, three term relation, special value, solving polynomial systems.