Some Properties of Hypergeometric Functions

ABSTRACT: This thesis consists of five chapters. The first chapter gives brief information about the thesis. In the second chapter, we give some preliminaries and auxilary results which we will use in thesis. In chapter three, the extension of beta function containing an extra parameter, which proved to be useful earlier, was used to extend Appell’s hypergeometric functions of two variables and extend Lauricella’s hypergeometric function of three variables. Furthermore, linear and bilinear generating relations for these extended hypergeometric functions are obtained by defining the extension of fractional derivative operator. Some properties of the extended fractional derivative operator are also presented. In chapter four, we consider generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new generalizations. In chapter five, we present various families of generating functions for a class of polynomials in two variables. Furthermore, several general classes of bilinear, bilateral or mixed multilateral generating functions are obtained for these polynomials. Keywords: Generating functions, Hypergeometric function, Fractional derivative operator, Gamma function, Beta function. ……………………………………………………………………………………………………………………………………………………………………………………………………………………