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

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## Abstract

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## 1. Introduction, Definitions and Preliminaries

#### 1.1. The Extended Beta, Gamma and Hypergeometric Functions

#### 1.2. The Riemann–Liouville Fractional Derivative Operator and Its Generalizations

#### 1.3. Extended Appell Hypergeometric Functions in Two Variables

## 2. The Generalized Extended Appell and Lauricella Functions

**Remark 1.**For various special choices of the sequence , the definitions in (2.2–2.4) would reduce to (known or new) extensions of the Gamma, Beta and hypergeometric functions. In particular, if we set

**Theorem 1.**For the generalized extended Appell functions

**Theorem 2.**For the functions

**Theorem 3.**For the functions

## 3. Applications of the Generalized Extended Riemann–Liouville Fractional Derivative Operator

**Remark 2.**The Definition (3.1) is easily recovered from (3.2) by specializing the sequence as in (2.5). Moreover, by using the specialization indicated in (1.9), the Definition (3.1) reduces immediately to (1.13). For , the Definitions (1.13), (3.1) and (3.2) would obviously reduce at once to the familiar Riemann–Liouville Definition (1.10). Each of these and the aforementioned other specializations are fairly straightforward. Henceforth, therefore, we choose to state our results in their general forms only and leave the specializations as an exercise for the interested reader.

**Theorem 4.**In terms of a suitably bounded multiple sequence let the multivariable function be defined by

**Remark 3.**The Beta function defined (for ) by

## 4. Mellin Transforms of the Generalized Extended Fractional Derivatives

**Theorem 5.**In terms of the generalized extended Gamma function defined by the Mellin transforms of the following generalized extended fractional derivatives defined by are given by

_{2}F

_{1}being the Gauss hypergeometric function.

## 5. A Set of Generating Functions

**Theorem 6.**Each of the following generating relations holds true for the generalized extended hypergeometric functions in one and more variables:

## 6. Concluding Remarks and Observations

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**MDPI and ACS Style**

Srivastava, H.M.; Parmar, R.K.; Chopra, P.
A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions. *Axioms* **2012**, *1*, 238-258.
https://doi.org/10.3390/axioms1030238

**AMA Style**

Srivastava HM, Parmar RK, Chopra P.
A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions. *Axioms*. 2012; 1(3):238-258.
https://doi.org/10.3390/axioms1030238

**Chicago/Turabian Style**

Srivastava, H. M., Rakesh K. Parmar, and Purnima Chopra.
2012. "A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions" *Axioms* 1, no. 3: 238-258.
https://doi.org/10.3390/axioms1030238