On the new fractional operators generating modified gamma and beta functions

Abstract

. In this paper, we introduce three new fractional operators, MRL fractional integral, MRL fractional derivative and MC fractional derivative operators, which including generalized M-series in their kernels and give some of their fundamental properties. Then we apply Laplace, Mellin and beta integral transformations to the new fractional operators and obtain conclusions involving classical gamma and beta and modified gamma and beta functions. As examples, we also obtain similar conclusions by applying new fractional operators to the functions z λ and ( 1 − az ) − λ . Furthermore, we present the relationships of the new fractional operators with other fractional operators in the literature. Finally, we compare the behavior of the function z 2 in classical Riemann-Liouville and Caputo fractional operators and MRL and MC fractional operators for orders ε = 0 . 2 , 0 . 4 , 0 . 6 , 0 . 8.