Rakesh K. Parmar

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

Extension of the fractional derivative operator of the Riemann-Liouville

By using the generalized beta function, we extend the fractional derivative operator of the Riemann-Liouville and discusses its properties. Moreover, we establish some relations to extended special functions of two and three variables via generating functions. c ©2017 All rights reserved.

Some Families of the Incomplete H-Functions and the Incomplete \(\overline H \)-Functions and Associated Integral Transforms and Operators of Fractional Calculus with Applications

Our present investigation is inspired by the recent interesting extensions (by Srivastava et al. [35]) of a pair of the Mellin–Barnes type contour integral representations of their incomplete generalized hypergeometric functions pγq and pΓq by means of the incomplete gamma functions γ(s, x) and Γ(s, x). Here, in this sequel, we introduce a family of the relatively more general incomplete H-functions γp,qm,n (z) and Γp,qm,n (z) as well as their such special cases as the incomplete Fox-Wright generalized hypergeometric functions pΨq(γ) [z] and pΨq(Γ) [z]. The main object of this paper is to study and investigate several interesting properties of these incomplete H-functions, including (for example) decomposition and reduction formulas, derivative formulas, various integral transforms, computational representations, and so on. We apply some substantially general Riemann–Liouville and Weyl type fractional integral operators to each of these incomplete H-functions. We indicate the easilyderivable extensions of the results presented here that hold for the corresponding incomplete

Fractional Integration and Differentiation of the Generalized Mathieu Series

\overline H

On an Extension of Extended Beta and Hypergeometric Functions

H¯-functions as well. Potential applications of many of these incomplete special functions involving (for example) probability theory are also indicated.

MATHIEU-TYPE SERIES BUILT BY (p, q)-EXTENDED GAUSSIAN HYPERGEOMETRIC FUNCTION

The main object of this paper is to introduce generalized Srivastava’s triple hypergeometric functions by using the generalized Pochhammer symbol and investigate certain properties, for example, their various integral representations, derivative formulas and recurrence relations. Various (known or new) special cases and consequences of the results presented here are also considered. c ©2017 All rights reserved.

On a Multivariable Class of Mittag-Leffler Type Functions

We aim to present some formulas for the Saigo hypergeometric fractional integral and differential operators involving the generalized Mathieu series S μ ( r ) , which are expressed in terms of the Hadamard product of the generalized Mathieu series S μ ( r ) and the Fox–Wright function p Ψ q ( z ) . Corresponding assertions for the classical Riemann–Liouville and Erdelyi–Kober fractional integral and differential operators are deduced. Further, it is emphasized that the results presented here, which are for a seemingly complicated series, can reveal their involved properties via the series of the two known functions.

Extended τ-hypergeometric functions and associated properties

Motivated mainly by certain interesting extensions of the -hypergeometric function defined by Virchenko et al. [11] and some -Appells function introduced by Al-Shammery and Kalla [1], we introduce here the -Lauricella functions , and and the confluent forms and of n variables. We then systematically investigate their various integral representations of each of these -Lauricella functions including their generating functions. Various (known or new) special cases and consequences of the results presented here are also considered.