Certain Generalizations of Quadratic Transformations of Hypergeometric and Generalized Hypergeometric Functions

There have been numerous investigations on the hypergeometric series 2F1 and the generalized hypergeometric series pFq such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extensions of one and several variables, continued fraction...

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Bibliographic Details
Published in:Symmetry (Basel) Vol. 14; no. 5; p. 1073
Main Authors: Qureshi, Mohd Idris, Choi, Junesang, Shah, Tafaz Rahman
Format: Journal Article
Language:English
Published: Basel MDPI AG 01.05.2022
Subjects:
Asymptotic series
Differential equations
Hypergeometric functions
Mathematical analysis
Transformations (mathematics)
ISSN:2073-8994, 2073-8994
Online Access:Get full text
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Summary:There have been numerous investigations on the hypergeometric series 2F1 and the generalized hypergeometric series pFq such as differential equations, integral representations, analytic continuations, asymptotic expansions, reduction cases, extensions of one and several variables, continued fractions, Riemann’s equation, group of the hypergeometric equation, summation, and transformation formulae. Among the various approaches to these functions, the transformation formulae for the hypergeometric series 2F1 and the generalized hypergeometric series pFq are significant, both in terms of applications and theory. The purpose of this paper is to establish a number of transformation formulae for pFq, whose particular cases would include Gauss’s and Kummer’s quadratic transformation formulae for 2F1, as well as their two extensions for 3F2, by making advantageous use of a recently introduced sequence and some techniques commonly used in dealing with pFq theory. The pFq function, which is the most significant function investigated in this study, exhibits natural symmetry.
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ISSN:2073-8994
2073-8994
DOI:10.3390/sym14051073