The Generalized $K$-Wright Function and Marichev-Saigo-Maeda Fractional Operators

In this paper, the generalized fractional operators involving Appell’s function $F_3$ in the kernel due to Marichev-Saigo-Maeda are applied to the generalized $K$-Wright function. These fractional operators when applied to power multipliers of the generalized $K$-Wright function ${}_p{\mathrm{\Psi }}_q^k$ yields a higher ordered generalized $K$-Wright function, namely, ${}_{p+3}{\mathrm{\Psi }}_{q+3}^k$. The Caputo-type modification of Marichev-Saigo-Maeda fractional differentiation is introduced and the corresponding assertions for Saigo and Erd'elyi-Kober fractional operators are also presented. The results derived in this paper generalize several recent results in the theory of special functions.