Generalized fractional calculus of the \(H\)-function associated with the Appell function \(F_3\) (Q2731631)

The object of this paper is to establish eight theorems for the generalized fractional integration and differentiation of arbitrary complex order for the \(H\)-function. The considered generalized fractional integration and differentiation operators contain the Appell function \(F_3\) as a kernel and are introduced recently by Saigo and Maeda. The operators studied in this paper generalize the classical Riemann-Liouville, Weyl, Erdélyi-Kober and Saigo operators. It has been shown that the generalized fractional integrals and derivatives of the \(H\)-function are transformed into other \(H\)-functions but of greater order. The results established provide extensions of the results given recently by Saigo and Kilbas. The derived results are more general than the known ones. Mellin transforms of the generalized fractional integrations are also discussed.