Some properties of Wright-type generalized hypergeometric function via fractional calculus

Abstract

This paper is devoted to the study of a Wright-type hypergeometric function (Virchenko, Kalla and Al-Zamel in Integral Transforms Spec. Funct. 12(1):89-100, 2001) by using a Riemann-Liouville type fractional integral, a differential operator and Lebesgue measurable real or complex-valued functions. The results obtained are useful in the theory of special functions where the Wright function occurs naturally. MSC: 33C20; 33E20; 26A33; 26A99

Generalized hypergeometric k-functions via (k,s)-fractional calculus

“…Setting s = 0 and k = 1 in (2.9) and (2.10) yields the known results (see [14]), which are recalled in the following corollary. Corollary 2.6.…”

“…The results presented here when s = 0 and (s, k) = (0, 1) reduce to some known identities in [8] and [14], respectively. Also, setting τ = 1 in the results here yields some corresponding identities involving the k-hypergeometric function 2 F 1,k (a, b; c; z).…”

“…Setting k = 1 and s = 0 in Definitions 2.2 and 2.3 reduces to the familiar Riemann-Liouville left-sided and right-sided fractional integrals and derivatives (see [14]). Theorem 2.4.…”

The INCOMPLETE GENERALIZED Τ-Hypergeometric AND SECOND Τ-Appell FUNCTIONS

“…Further, we have introduced and investigated the family of incomplete second τ -Appell hypergeometric functions Γ τ1,τ2 2 and γ τ1,τ2 2 of two variables. The special cases of the results presented here when κ = 0 would reduce to the corresponding well-known results for the generalized τ -hypergeometric function (see, for details, [10,12,18,24,41,42]) and second τ -Appell function (see, for details, [2]). …”

“…The special cases of (2.23)-(2.25) when τ = 1 yield the corresponding known relations for the generalized τ -hypergeometric function [24].…”