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

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