On the new fractional operators generating modified gamma and beta functions

Enes Ata;


In this paper, three new fractional operators, MRL fractional integral, MRL fractional derivative and MC fractional derivative operators, which including generalized M-series at their kernels, are defined and their obtained to be linear. Then, by applying Laplace, Mellin and beta integral transforms to the new fractional operators, results including classical gamma, classical beta, modified gamma and modified beta functions are obtained. Also, as examples, again similar results are obtained by applying the new fractional operators to functions $z^{\lambda}$ and $(1-az)^{-\lambda}$. After, the relationships of the new fractional operators with other fractional operators in the literature are presented. Finally, the behavior of the function $z^{2}$ in classical fractional operators (Riemann-Liouville and Caputo fractional operators) of orders $\varepsilon=0.2,0.4,0.6,0.8$ and new fractional operators (MRL and MC fractional operators) of orders $\varepsilon=0.2,0.4,0.6,0.8$ is compared.

Vol. 24 (2023), No. 3, pp. 1127-1144
DOI: 10.18514/MMN.2023.4124

Download: MMN-4124