On computing some special values of hypergeometric functions

TL;DR

This paper advances the computation of special values of hypergeometric functions, which are important in mathematics and physics, by generalizing identities for multivariable cases using two different approaches.

Contribution

It introduces new formulas for multivariable hypergeometric functions by extending Kummer's identity, enhancing their computability inside and outside the convergence disk.

Findings

New formulas for multivariable hypergeometric functions

Generalization of Kummer's identity to multiple variables

Improved methods for computing special hypergeometric values

Abstract

The theoretical computing of special values assumed by the hypergeometric functions has a high interest not only on its own, but also in sight of the remarkable implications to both pure Mathematics and Mathematical Physics. Accordingly, in this paper we continue the path of research started in two our previous papers appeared on [30] and [31] providing some contribution to such functions computability inside and outside their disk of convergence. This is accomplished through two different approaches. The first is to provide some new results in the spirit of theorem 3.1 of 31] obtaining formulae for multivariable hypergeometric functions by generalizing a well known Kummer's identity to the hypergeometric functions of two or more variable like those of Appell and Lauricella.