Abstract

This work is focused on the study of the generalized forms of the fractional derivatives of Reimann-Liouville, Caputo and Hilfer, in terms of Ψ functions. The fractional derivatives of Ψ-Prabhakar, Ψ-Hilfer-Prabhakar, and its regularized form are also described in terms of Ψ-Mittage-Leffler type functions. These are then used to solve a number of Cauchy type equations involving Ψ-Hilfer-Prabhakar fractional derivatives and their regularized form, including the generalized fractional free electron laser equation.