Some results on the extended beta and extended hypergeometric functions

A New Extension of Extended Caputo Fractional Derivative Operator

“…Recently, many papers (see [1,8,12,16,21]) introduced various extension and generalization of fractional calculus . In this section, we define a new extension of extended Caputo fractional derivative operator.…”

Some Integral Transforms Involving Extended Generalized Gauss Hypergeometric Functions

“…Virchenko and Ovcharenko [16] presented some new integral transforms with the generalized confluent hypergeometric function due to Virchenko [15]. Here, in this paper, using the extended generalized integral transform given by Luo et al [6], we introduce some new generalized integral transforms to investigate such their (potentially) useful properties as inversion formulas and Parseval-Goldstein type relations. Classical integral transforms including (for example) Laplace, Stieltjes, and Widder-Potential transforms are seen to follow as special cases of the newly-introduced integral transforms.…”

“…It is also noted that for p = 2 and q = 1 the definitions in (1.1) would reduce immediately to the extended hypergeometric type function defined as follows (see [6]):…”

“…Recently, Luo et al [6] introduced the following extended generalized hypergeometric function p F (α,β;κ,µ) q and investigated its various properties. The extended generalized hypergeometric function p F (α,β;κ,µ) q is defined by…”

Upper and lower bounds of integral operator defined by the fractional hypergeometric function

“…Various properties of the function (2) have been studied by Luo et al (2014) [23]. The special case of (2) when p D 0 is seen to immediately reduce to the familiar beta function B.x; y/ (min f<.x/; <.y/g > 0) (see, e.g., Srivastava et al [24]).…”