Generalized fractional Hilfer integral and derivative

Fractional calculus, a branch of mathematics, is focused on the study and applications of the differential and integral operators of non-integer order. Although the fractional calculus is as old as the classical calculus, it has become one of the most developed areas of mathematics only in the last 40 years, not only because of the exponential growth of the number of publications in this area, but also due to its different applications and its overlapping with other areas of mathematics. This area has been developed intensively in recent years and it has found multiple applications in various fields. The classical results were basically extended in two fundamental directions: Riemann–Liouville fractional derivative and Caputo fractional derivative. As a result of the progress made in this area, numerous fractional (global) and generalized (local) operators have been appeared. These new operators give researchers the possibility to choose the one that suits best with the problem they investigate. Readers can consult the paper [2] where some reasons are given to justify the appearance of these new operators and where the applications and theoretical developments of these operators are discussed. These operators, developed by many mathematicians with a hardly specific formulation, include the Riemann–Liouville (RL), Weyl, Erdelyi-Kober and Hadamard integrals, and the fractional operators of Liouville and Katugampola. Many authors have even introduced new fractional operators generated from the general local differential operators. In this direction, a generalized local derivative was defined in [19], which generalizes both the conformable and non-conformable derivatives and that is the basis for the generalized integral operator proposed in [7], which contains as a particular case the fractional integral of Riemann– Liouville (see [31]). In fact, these new operators require a classification as they can cause confusion in researchers. Baleanu and Fernandez [3] gave a fairly complete classification of these fractional and generalized operators together with abundant information and references. For a more complete review, the readers are referred to Chapter 1 of [1], where a history of differential operators (both local and global) from Newton to Caputo is presented and where the qualitative differences between the operators are shown. Section 1.4 of [1] contains some conclusions that we want to highlight: “Therefore, we can conclude that the Riemann–Liouville and Caputo operators are not derivatives and, therefore, they are not fractional derivatives, but fractional operators. We agree with the result [27] that the local fractional operator is not a fractional derivative” (see p.24 in [1]). In this work, we present a new definition of the k-generalized fractional derivative of the Hilfer type, and we study its fundamental properties. We also present a particular case with a kernel defined in terms of the sigmoid function. The gamma function Γ (see [21,24,28,29]) and k-generalized gamma function Γk (see [6]) are defined as: