Generalized Erdélyi-Kober Fractional Integrals and Images of Special Functions

The Riemann–Liuoville fractional integrals are the simplest and most popular operators of the classical fractional calculus. But their variants, the Erdélyi–Kober operators of fractional integration, have many more applications due to the freedom to choose the additional (three) parameters. We introduce and study a generalization of the Erdélyi–Kober and Riemann–Liuoville fractional integrals, where the elementary kernel function is replaced by a suitably chosen I1,11,0-function. The I-functions introduced by Rathie in 1997 are generalized hypergeometric functions extending the Fox H-functions and the Meijer G-functions. Note that till recently this new class of special functions has not been popular because of their too complicated structure involving fractional powers of the Gamma functions and their multi-valued behavior. However, the I-functions happened to arise not only for the needs of statistical physics, but also since they included important special functions in mathematics that were not covered by the H- and G-functions. In our previous works, as Kiryakova and Paneva-Konovska, we have shown the relations of such functions, among which are the Mittag–Leffler and Le Roy type, their multi-index variants, and others related to fractional calculus, to the I-functions. Here, we propose a new theory of generalization of the Erdélyi–Kober fractional integrals, based on the use of an I-function as a kernel. This will serve next as a base to extend our generalized multi-order fractional calculus with operators involving Im,mm,0. In this paper, we also evaluate the images under these new generalized fractional integrals of special functions of very general form. Finally, in the Conclusion section, we comment on some earlier discussions on the relations between fractal geometry and fractional calculus, nowadays already without any doubts.