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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2021
• Main Authors: Çetinkaya, Asena, Karp, Dmitrii, Prilepkina, Elena
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2021
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211429
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Hypergeometric Functions at Unit Argument: Simple Derivation of Old and New Identities. Asena Çetinkaya, Dmitrii Karp and Elena Prilepkina. SIGMA 17 (2021), 098, 25 pages

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 function. For instance, we recover two- and three-term Thomae relations for ₃ ₂, give two- and three-term transformations for ₄ ₃ with one unit shift and ₅ ₄ with two unit shifts in the parameters, establish multi-term identities for general ₚ ₚ₋₁ and several transformations for terminating Kampé de Fériet and Srivastava ⁽³⁾ functions. We further present a presumably new formula for analytic continuation of 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 -hypergeometric functions to derive multi-term relations for terminating series.