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

Abstract

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.