Certain Integral Transform and Fractional Integral Formulas for the Generalized Gauss Hypergeometric Functions

Junesang Choi,Praveen Agarwal

doi:10.1155/2014/735946

Abstract

A remarkably large number of integral transforms and fractional integral formulas involving various special functions have been investigated by many authors. Very recently, Agarwal gave some integral transforms and fractional integral formulas involving theFp(α,β)(·). In this sequel, using the same technique, we establish certain integral transforms and fractional integral formulas for the generalized Gauss hypergeometric functionsFp(α,β,m)(·). Some interesting special cases of our main results are also considered.

Highlights

Introduction and PreliminariesThe theory of special functions has been one of the most rapidly growing research subjects in mathematical analysis
Agarwal [11] gave some interesting integral transform and fractional integral formulas involving (7). In this sequel, using the same technique, we propose to derive some integral transforms and image formulas for the generalized Gauss hypergeometric function (8) by applying certain integral transforms and general pair of fractional integral operators involving Gauss hypergeometric function 2F1, which will be introduced in Sections 2 and 3, respectively
The Saigo fractional integrations of generalized Gauss hypergeometric type functions (8) are given by the following results

Summary

Introduction and Preliminaries

The theory of special functions has been one of the most rapidly growing research subjects in mathematical analysis. It is easy to see that the special case of (1) when m = 1 reduces to the well-known generalized beta type function defined by (see, e.g., [7, page 4602, Equation (4)]; see [6, page 32, Chapter 4]). By appealing to Bp(α,α)(x, y), Ö zergin et al introduced and investigated a further extension of the following potentially useful generalized Gauss hypergeometric functions defined as follows (see, e.g., [7, page 4606, Section 3]; see [6, page 39, Chapter 4]): Fp(α,β) (a, b; c; z) = ∑. By using the more generalized beta function (1), Parmar [9] introduced and investigated a family of the following potentially useful generalized Gauss hypergeometric functions defined as follows (see [9, page 44]): Fp(α,β;m) (a, b; c; z) = ∑. We consider some interesting special cases of our main results

Integral Transform and the Generalized Gauss Hypergeometric Functions

Fractional Calculus of the Generalized Gauss Hypergeometric Functions

Concluding Remarks