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

Abstract

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 ${}_{p}\Psi _{q}^{(\gamma )}(\mathfrak{t})$ Ψ q ( γ ) p ( t ) and ${}_{p}\Psi _{q}^{(\Gamma )}(\mathfrak{t})$ Ψ q ( Γ ) p ( 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.

Certain Integral and Differential Formulas Involving the Product of Srivastava’s Polynomials and Extended Wright Function

“…Following are the special cases of results discussed above. (i) The formulas obtained reduce to formulas for Saigo's fractional integral operators ( 16) and (17) for ξ = 0; (ii) By substituting η = −ξ in Saigo's fractional integral operators, we obtain the formulas for Riemann-Liouville fractional integral operators (20) and (21).…”

“…Suthar established the composition formulae for the k-fractional calculus operators associated with k-Wright function [20]. Fractional calculus and integral transforms of general class of polynomials and incomplete Fox-Wright functions is discussed by Jangid et al [21]. Certain expressions of the Laguerre polynomial and relations of some known functions in terms of generalized Meijer G-functions are explored in [22,23].…”