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

H M Srivastava,Rakesh K Parmar,Purnima Chopra

doi:10.3390/axioms1030238

Abstract

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.

Highlights

In the third subsection (Part 1.3), we introduce the extended Appell hypergeometric functions in two variables, which were recently investigated in conjunction with the family of the extended Riemann–Liouville fractional derivative operators defined in Part 1.2
In general, the natural connection of the Riemann–Liouville fractional derivative operator Dzμ defined by (1.10) with ordinary derivatives when the order μ is zero or a positive integer is lost by the extended fractional derivative operator in Definition (1.13) and its further generalizations which we have considered in our present investigation
The aim of this paper is to investigate the various properties of a further generalization of the extended fractional derivative operator Dzμ,p defined by (1.13) and apply the generalized operator to derive generating relations for hypergeometric functions in one, two and more variables

Summary

The Riemann–Liouville Fractional Derivative Operator and Its Generalizations

For the Riemann–Liouville fractional derivative operator Dzμ defined by The path of integration in the definition (1.10) is a line in the complex t-plane from t = 0 to t = z. (1.2), Özarslan and Özergin [1] defined the correspondingly extended Riemann–Liouville fractional derivative operator Dzμ,p by. The path of integration in the Definition (1.13), which immediately yields the definition (1.10) when p = 0, is a line in the complex t-plane from t = 0 to t = z. In general, the natural connection of the Riemann–Liouville fractional derivative operator Dzμ defined by (1.10) with ordinary derivatives when the order μ is zero or a positive integer is lost by the extended fractional derivative operator in Definition (1.13) and its further generalizations which we have considered in our present investigation

Extended Appell Hypergeometric Functions in Two Variables

The Generalized Extended Appell and Lauricella Functions

Mellin Transforms of the Generalized Extended Fractional Derivatives

A Set of Generating Functions

Concluding Remarks and Observations