Composition Formulae for the k-Fractional Calculus Operator with the S-Function

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.