Certain Fractional Integral and Differential Formulas Involving the Extended Incomplete Generalized Hypergeometric Functions

Abstract

The fractional integral and differential operators involving the family of special functions have found significant importance and applications in various fields of mathematics and engineering. The goal of this chapter is to find the fractional integral and differential formulas (also known as composition formulas) involving the extended incomplete generalized hypergeometric functions by using the generalized fractional calculus operators (the Marichev–Saigo–Maeda operators). After that, we established their image formulas by using the integral transforms like: Beta transform, Laplace transform and Whittaker transform. Moreover, the reduction formulas are also considered as special cases of our main findings associated with the well-known Saigo fractional integral and differential operators, Erdelyi-Kober fractional integral and differential operators, Riemann-Liouville fractional integral and differential operators and the Weyl fractional calculus operators.