Hypergeometric functions at unit argument: simple derivation of old and new identities

A. Çetinkaya, D. B. Karp, E. G. Prilepkina

Published 2021-05-11, updated 2021-05-19Version 2

The main goal of this paper is to derive a number of identities for 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 ${}_3F_2$, give two- and three- term transformations for ${}_4F_3$ with one unit shift and ${}_5F_4$ with two unit shifts in the parameters, establish multi-term identities for ${}_pF_{p-1}$ and several transformations for terminating Kamp\'e de F\'eriet and Srivastava hypergeometric functions of two and three variables. We further present a presumably new formula for analytic continuation in parameters for generalized hypergeometric function evaluated at unity and reveal somewhat unexpected connections between the generalized hypergeometric functions and generalized and ordinary Bernoulli polynomials. Finally, we exploit some recent duality relations for the generalized hypergeometric and q-hypergeometric functions to derive multi-term identities for terminating series.