Hypergeometric Differential Equation and New Identities for the Coefficients of N{\o}rlund and B\"uhring

TL;DR

This paper reviews existing results on hypergeometric functions near unity, then introduces new formulas and identities for coefficients in their expansions, connecting classical functions with Meijer's G-function.

Contribution

It provides new identities for coefficients in hypergeometric and Meijer's G-function expansions, linking classical and modern special functions.

Findings

New formulas for expansion coefficients of hypergeometric functions.

Identities connecting hypergeometric coefficients with Meijer's G-function.

Revealing forgotten identities such as Hermite's sine function identity.

Abstract

The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by N{\o}rlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by B\"uhring, Srivastava and Saigo. In the first part of this paper we review their results rewriting them in terms of Meijer's $G$-function and explaining the interconnections between them. In the second part we present new formulas and identities for the coefficients that appear in the expansions of Meijer's $G$-function and generalized hypergeometric function around unity. Particular cases of these identities include known and new relations for Thomae's hypergeometric function and forgotten Hermite's identity for the sine function.