Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 098, 25 pages      arXiv:2105.05196      https://doi.org/10.3842/SIGMA.2021.098

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

Asena Çetinkaya a, Dmitrii Karp bc and Elena Prilepkina cd
a) İstanbul Kultur University, İstanbul, Turkey
b) Holon Institute of Technology, Holon, Israel
c) Far Eastern Federal University, Ajax Bay 10, Vladivostok, 690922, Russia
d) Institute of Applied Mathematics, FEBRAS, 7 Radio Street, Vladivostok, 690041, Russia

Received May 20, 2021, in final form October 31, 2021; Published online November 07, 2021

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 ${}_3F_2$, 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 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 ${}_pF_{p-1}(1)$ 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.

Key words: generalized hypergeometric function; Meijer's $G$ function; multiple hypergeometric series; Kampé de Fériet function; Srivastava function; hypergeometric identity; generalized Bernoulli polynomials.

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