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

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## Abstract

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 _{ p }γ_{ q } and _{ p }Γ_{ q } 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 γ _{ p,q } ^{ m,n } (*z*) and Γ _{ p,q } ^{ m,n } (*z*) as well as their such special cases as the incomplete Fox-Wright generalized hypergeometric functions _{ p }Ψ _{ q } ^{(γ)} [*z*] and _{ p }Ψ _{ q } ^{(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 \(\overline H \)-functions as well. Potential applications of many of these incomplete special functions involving (for example) probability theory are also indicated.

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