Generalization of Special Functions and Explicit Form of Fractional Derivative of Rational Functions

Abstract

The goal of this paper is to extend the classical fractional derivatives. For this purpose, it is introduced the new extended modified Bessel function and also given an important relation between this new function Iυ(q; x) and the confluent hypergeometric function 1F1(α, β, x). Besides, it is used to generalize the hypergeometric, the confluent hypergeometric and the extended beta functions by using the new extended modified Bessel function. Also, the asymptotic formulae and the generating function of the extended modified Bessel function are obtained. The extensions of classical fractional derivatives are defined via extended modified Bessel function and, first time the fractional derivative of rational functions is explicitly given via complex partial fraction decomposition.