Integral transform and fractional derivative formulas involving the extended generalized hypergeometric functions and probability distributions

Abstract

In the present paper, our aim is to establish several formulas involving integral transforms, fractional derivatives, and a certain family of extended generalized hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results. A probability density function involving the extended generalized hypergeometric function is introduced, and its properties are studied. The corresponding properties of some of the classical probability distributions and their associated probability density functions are easily derivable as special cases of our general results. Copyright © 2016 John Wiley & Sons, Ltd.

Certain fractional calculus formulas involving extended generalized Mathieu series

“…For various investigations involving the Hadamard product (or the convolution), the interested reader may refer to recent papers on the subject (see, for example, [68,69] and the references cited therein). Also we require the Fox-Wright function p q (z) (p, q ∈ N 0 ) with p numerator and q denominator parameters defined for a 1 , .…”

Fractional Integration and Differentiation of the Generalized Mathieu Series

“…For various other investigations involving the Hadamard product (or the convolution), the interested reader may be referred to several recent papers on the subject (see, for example, [27,28] and the references cited in each of these papers).…”

A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas

“…Agarwal et al [2] made use of the P δ -transform given by Definition 2 in order to solve a fractional Volterra type integral equation and a non-homogeneous time-fractional heat equation involving a so-called pathway-type integral transform which is, in fact, the same as the extensively-and widely-investigated Riemann-Liouville fractional integral with, of course, some obvious parameter and variable changes. On the other hand, Srivastava et al [25] found many results involving a family of generalized hypergeometric functions by using the P δ -transform given by Definition 2.…”

“…The Jacobi transform of the power function z ρ−1 (see, for example, [25,Eq. (20)]) is given by (18) yields the following well-known result for the Jacobi transform of z m (m ∈ N 0 ), which is given by (see, for example, [20,p.…”