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Motivated by a recent study on certain families of the incomplete H -functions (Srivastava et al. in Russ. J. Math. Phys. 25(1):116–138, 2018 ), we aim to investigate and develop several interesting properties related to product of a more general polynomial class together with incomplete Fox–Wright hypergeometric functions Ψ q ( γ ) p ( t ) ${}_{p}\Psi _{q}^{(\gamma )}(\mathfrak{t})$ and Ψ q ( Γ ) p ( t ) ${}_{p}\Psi _{q}^{(\Gamma )}(\mathfrak{t})$ including Marichev–Saigo–Maeda (M–S–M) fractional integral and differential operators, which contain Saigo hypergeometric, Riemann–Liouville, and Erdélyi–Kober fractional operators as particular cases regarding different parameter selection. Furthermore, we derive several integral transforms such as Jacobi, Gegenbauer (or ultraspherical), Legendre, Laplace, Mellin, Hankel, and Euler’s beta transforms.

Fractional calculus and integral transforms of the product of a general class of polynomial and incomplete Fox–Wright functions