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Two fractional integral operators associated with FoxH-function due to Saxena and Kumbhat are applied toM-series, which is an extension of both Mittag-Leffler function and generalized hypergeometric functionpFq. The Mellin and Whittaker transforms are obtained for these compositional operators withM-series. Further some interesting properties have been established including power function and Riemann-Liouville fractional integral operators. The results are expressed in terms ofH-function, which are in compact form suitable for numerical computation. Special cases of the results are also pointed out in the form of lemmas and corollaries.

In this study, the S-function is applied to Saigo’s k -fractional order integral and derivative operators involving the k -hypergeometric function in the kernel; outcomes are described in terms of the k -Wright function, which is used to represent image formulas of integral transformations such as the beta transform. Several special cases, such as the fractional calculus operator and the S -function, are also listed.

The main object of this paper is to present certain new image formulas for the product of general class of polynomial and (p, q)-extended Gauss's hypergeometric function by applying the Saigo-Maeda fractional integral operators involving Appell's function F3. Certain interesting special cases of our main results are also considered.