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 ${}_3{F}_2$, give two- and three-term transformations for ${}_4{F}_3$ with one unit shift and ${}_5{F}_4$ with two unit shifts in the parameters, establish multi-term identities for general ${}_p{F}_{p-1}$ and several transformations for terminating Kampé de Fériet and Srivastava $F^{(3)}$ functions. We further present a presumably new formula for analytic continuation of ${}_p{F}_{p-1}(1)$ 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.