Hypergeometric Differential Equation and New Identities for the Coefficients of Nørlund and Bühring

Abstract

The fundamental set of solutions of the generalized hypergeometric differential equation in the neighborhood of unity has been built by Nørlund in 1955. The behavior of the generalized hypergeometric function in the neighborhood of unity has been described in the beginning of 1990s by Bühring, 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.

References in corpus (4)

• Analytic Continuation of Hypergeometric Functions in the Resonant Case
• Completely monotonic gamma ratio and infinitely divisible H-function of Fox
• Products of Beta distributed random variables
• A short proof of duality relations for hypergeometric functions

Cited by in corpus (5)

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• Beyond the beta integral method: transformation formulas for hypergeometric functions via Meijer's G function
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• Hypergeometric Functions at Unit Argument: Simple Derivation of Old and New Identities
• Further applications of the G function integral method