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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 …