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

• Опубліковано в: : Symmetry, Integrability and Geometry: Methods and Applications
• Дата: 2021
• Автори: Çetinkaya, Asena, Karp, Dmitrii, Prilepkina, Elena
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Інститут математики НАН України 2021
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/211429
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: 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 ₃₂, give two- and three-term transformations for ₄₃ with one unit shift and ₅₄ with two unit shifts in the parameters, establish multi-term identities for general ₚₚ₋₁ and several transformations for terminating Kampé de Fériet and Srivastava ⁽³⁾ functions. We further present a presumably new formula for analytic continuation of ₚₚ₋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.