# Generalization of Special Functions and its Applications to Multiplicative and Ordinary Fractional Derivatives

## TL;DR

This paper introduces a new extended modified Bessel function to generalize classical and multiplicative fractional derivatives, establishing relations with hypergeometric functions and deriving explicit formulas for fractional derivatives of rational functions.

## Contribution

It presents a novel extended modified Bessel function and uses it to generalize various special functions and fractional derivatives, including explicit formulas for rational functions.

## Key findings

- New extended modified Bessel function and its properties
- Generalizations of hypergeometric and beta functions
- Explicit fractional derivatives of rational functions

## Abstract

The goal of this paper is to extend the classical and multiplicative fractional derivatives. For this purpose, it is introduced the new extended modified Bessel function and also given an important relation between this new function I(v,q;x) and the confluent hypergeometric function. 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 and multiplicative 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.