A new generalized class of fractional operators with weight and respect to another function

Abstract

This paper introduces and investigates the properties of a new generalized class of fractional differential and integral operators. Such newly class covers various definitions of fractional derivatives with singular and non-singular kernels, weighted fractional derivatives with respect to another function, as well as the new mixed fractional derivative in the sense of Caputo and Riemann-Liouville. Furthermore, the newly introduced class includes all existing forms of fractional integrals, weighted fractional
integrals and also the weighted fractional integrals with respect to another function including Riemann-Liouville, Hadamard, Katugampola and Hattaf fractional integrals. Moreover, some fundamental properties of the new generalized class of fractional differential and integral operators are rigorously derived.