Certain Generalizations of Quadratic Transformations of Hypergeometric and Generalized Hypergeometric Functions.

There have been numerous investigations on the hypergeometric series 2 F 1 and the generalized hypergeometric series p F q 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 2 F 1 and the generalized hypergeometric series p F q are significant, both in terms of applications and theory. The purpose of this paper is to establish a number of transformation formulae for p F q , whose particular cases would include Gauss's and Kummer's quadratic transformation formulae for 2 F 1 , as well as their two extensions for 3 F 2 , by making advantageous use of a recently introduced sequence and some techniques commonly used in dealing with p F q theory. The p F q function, which is the most significant function investigated in this study, exhibits natural symmetry. [ABSTRACT FROM AUTHOR]