Hypergeometric Functions at Unit Argument: Simple Derivation of Old and New Identities

Abstract

The main goal of this paper is to derive a number of identities for the generalized hypergeometric function evaluated at unity and for certain terminating multivariate hypergeometric functions from the symmetries and other properties of Meijer's 𝐺 function. For instance, we recover two- and three-term Thomae relations for ₃𝐹₂, give two- and three-term transformations for ₄𝐹₃ with one unit shift and ₅𝐹₄ with two unit shifts in the parameters, establish multi-term identities for general ₚ𝐹ₚ₋₁ and several transformations for terminating Kampé de Fériet and Srivastava 𝐹⁽³⁾ functions. We further present a presumably new formula for analytic continuation of ₚ𝐹ₚ₋₁(1) in parameters and reveal somewhat unexpected connections between the generalized hypergeometric functions and the generalized and ordinary Bernoulli polynomials. Finally, we exploit some recent duality relations for the generalized hypergeometric and 𝑞-hypergeometric functions to derive multi-term relations for terminating series.