Generalized Extended Riemann-Liouville type fractional derivative operator

TL;DR

This paper introduces new generalized fractional derivative operators based on extended special functions, providing recurrence relations, transformations, and integral representations to advance fractional calculus theory.

Contribution

It presents a novel extension of the Riemann-Liouville fractional derivative using generalized special functions and derives related mathematical properties.

Findings

New generalized fractional derivative operator introduced

Recurrence relations and transformation formulas derived

Integral representations established

Abstract

In this paper, we aim to present new extensions of incomplete gamma, beta, Gauss hypergeometric, confluent hypergeometric function and Appell-Lauricella hypergeometric functions, by using the extended Bessel function due to Boudjelkha [4]. Some recurrence relations, transformation formulas, Mellin transform and integral representations are obtained for these generalizations. Further, an extension of the Riemann-Liouville fractional derivative operator is established.