Siago’s K-fractional calculus operators

Abstract

The aim of present paper is to define a pair of $k$-Saigo fractional integral and derivative operators involving generalized $k$-hypergeometric function. The Saigo-k generalized fractional operators involving $k$-hypergeometric function in the kernel are applied to the generalized $k$-Mittag-Leffler function and evaluate the formula
$$
{ }_2 F_{1, k}\left[\begin{array}{cc}
(\alpha, k),(\beta, k) & \\
(\gamma, k) & ; \frac{1}{k}
\end{array}\right]=\frac{\Gamma_k(\gamma) \Gamma_k(\gamma-\alpha-\beta)}{\Gamma_k(\gamma-\alpha) \Gamma_k(\gamma-\beta)}
$$
using the integral representation for $k$-hypergeometric function.