On a $$\psi $$-Generalized Fractional Derivative Operator of Riemann–Liouville with Some Applications

Abstract

Various extended beta functions have been investigated by many researchers (Mubeen et al. in FJMS 102:1545–1557, 2017; Ozergin et al. in J Comput Appl Math 235:4601–4610, 2011; Parmar et al. in J Class Anal 11:91–106, 2017). Our aim in this paper is to develop $$\psi $$ -generalized Riemann Liouville fractional derivative operator by using $$\psi $$ -generalized beta function. We also present some new results on $$\psi $$ -generalized beta function and introduce some integral representations of $$\psi $$ -generalized hypergeometric functions. Moreover, we establish fractional derivative formulas of some known functions and discuss the properties of Mellin transform. Further we discuss some applications based on generating relation for the $$\psi $$ -generalized hypergeometric functions with the help of new $$\psi $$ -generalized Riemann–Liouville fractional derivative operator.