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 hyper-geometric functions from the symmetries and other properties of Meijer's G function. For instance, we recover two- and three-term Thomae relations for F-3(2), give two- and three-term transformations for F-4(3) with one unit shift and F-5(4) with two unit shifts in the parameters, establish multi-term identities for general F-p(p-1) and several transformations for terminating Kampe de Feriet and Srivastava F-(3) functions. We further present a presumably new formula for analytic continuation of F-p(p-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.