Generalized Leibniz Rule for an Extended Fractional Derivative Operator with Applications to Special Functions

Recently an extended operator of fractional derivative related to a generalized beta function has been used in order to obtain some generating relations involving extended hypergeometric functions. In this paper, an extended fractional derivative operator with respect to an arbitrary regular and univalent function based on the Cauchy integral formula is defined. This is done to compute the extended fractional derivative of the function log z and principally, to obtain a generalized Leibniz rule. Some examples involving special functions are given. A representation of the extended fractional derivative operator in terms of the classical fractional derivative operator is also determined by using a result of A.R. Miller.