The Multi-Index Mittag-Leffler-Le Roy Functions as I- and H¯-Functions and New Fractional Calculus Operators

Abstract

In the last decade, the interest in a special function named after the French mathematician Le Roy has provoked several authors to introduce and study its various extensions. Historically, the Le Roy function appeared almost in same time and in similar way of goals like the Mittag-Leffler function that has important role in Fractional Calculus. The Le Roy type functions are also “fractional exponentials” but the fractionalizing parameters appear as power indices of the involved Gamma functions. In our recent works, we have introduced rather general multi-index Le Roy type functions, studied their analytical properties and proposed their association to the class of Special Functions of Fractional Calculus.A newly developed idea is to show that the Le Roy type functions can be represented in terms of the I-functions of Rathie and H-functions of Inayat-Hussain that are further extensions of the Fox H¯-functions and Fox-Wright pΨq-functions. It happens that other important mathematical functions as the polylogarithms, Riemann Zeta function and its extensions, also belong to this more general class of special functions. We have solved, at least in some simpler case, the problem to find integral and differential-like operators for which the Le Roy type functions are eigenfunctions. This lead us to a new class of operators with I-functions kernels that can be considered as further extensions of the operators of generalized fractional calculus.