Extended hypergeometric and confluent hypergeometric functions

Recently several authors have extended the Gamma function, Beta function, the hypergeometric function, and the con-fluent hypergeometric function by using their integral representa-tions and provided many interesting properties of their extended functions. Here we aim at giving further extensions of the above-mentioned extended functions and investigating various formulas for the further extended functions in a systematic manner. More-over, our extension of the Beta function is shown to be applied to Statistics and also our extensions find some connections with other special functions and polynomials such as Laguerre polynomials, Macdonald and Whittaker functions.

Extension of extended beta, hypergeometric and confluent hypergeometric functions

Some generating relations for extended hypergeometric functions via generalized fractional derivative operator

Generating functions for the generalized Gauss hypergeometric functions

Recently, an extended operator of fractional derivative related to a generalized Beta function was used in order to obtain some generating relations involving the extended hypergeometric functions [1]. The main object of this paper is to present a further generalization of the extended fractional derivative operator and apply the generalized extended fractional derivative operator to derive linear and bilinear generating relations for the generalized extended Gauss, Appell and Lauricella hypergeometric functions in one, two and more variables. Some other properties and relationships involving the Mellin transforms and the generalized extended fractional derivative operator are also given.

https://www.mdpi.com/2075-1680/1/3/238/pdf

A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions

Extension of Euler's beta function

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Extension of Euler's beta function