On a solution of a fractional hyper-Bessel differential equation by means of a multi-index special function

TL;DR

This paper introduces a multi-parameter extension of the Wright function derived from a hyper-Bessel operator with fractional derivatives, providing new solutions to fractional differential equations and connecting to known special functions.

Contribution

It develops a novel multi-index special function generalizing Wright and Mittag-Leffler functions, linked to hyper-Bessel operators with Caputo fractional derivatives.

Findings

The new function generalizes classical special functions.

It provides isochronous solutions to nonlinear fractional PDEs.

Connections to known functions like Mittag-Leffler are established.

Abstract

In this paper, we introduce a new multiple-parameters (multi-index) extension of the Wright function that arises from an eigenvalue problem for a case of hyper-Bessel operator involving Caputo fractional derivatives. We show that by giving particular values to the parameters involved in this special function, this leads to some known special functions (as the classical Wright function, the $\alpha$-Mittag-Leffer function, the Tricomi function, etc.) that on their turn appear as cases of the so-called multi-index Mittag-Leffer functions. As an application, we mention that this new generalization Wright function is an isochronous solution of a nonlinear fractional partial differential equation.