Bulletin of the
Korean Mathematical Society BKMS

Fractional calculus operators have been investigated by many authors during the last four decades due to their importance and usefulness in many branches of science, engineering, technology, earth sciences and so on. Saigo \emph{et al}. \cite{SaigoMaeda1996} evaluated the fractional integrals of the product of Appell function of the third kernel $F_3$ and multivariable $H$-function. In this sequel, we aim at deriving the generalized fractional differentiation of the product of Appell function $F_3$ and multivariable $H$-function. Since the results derived here are of general character, several known and (presumably) new results for the various operators of fractional differentiation, for example, Riemann-Liouville, Erd\'{e}lyi-Kober and Saigo operators, associated with multivariable $H$-function and Appell function $F_3$ are shown to be deduced as special cases of our findings.

Keywords: multivariable $H$-function, Saigo fractional calculus operators, Saigo-Maeda operators, fractional calculus, Appell function $F_{3}$, $H$-function, Riemann-Liouville derivative operator

MSC numbers: Primary 26A33; Secondary 33C45