A STUDY OF GENERALIZED APPELL'S FUNCTIONS & RL FRACTIONAL DERIVATIVE OPERATOR

In this work, we introduce a novel and intriguing generalized version of Appell's functions in conjunction with the Riemann-Liouville fractional derivative operator. By exploring the analytic properties and applications of this new Riemann–Liouville type fractional derivative operator, we have derived new formulae for the fractional derivatives of various well-known functions. These are expressed in terms of newly extended Appell's two variables hypergeometric functions and Lauricella's three variables hypergeometric functions. Additionally, we have defined the Mellin transformations of these functions. We have formulated some generating functions for generalized extended hypergeometric functions using our new definitions of extended Appell's functions and the extended Riemann-Liouville fractional derivative operator to further substantiate our new operator