A note on Hadamard fractional differential equations with varying coefficients and their applications in probability

TL;DR

This paper explores the relationship between Hadamard fractional differential equations with varying coefficients, special functions from generalized COM-Poisson distributions, and their applications in probability, revealing new analytical insights.

Contribution

It introduces new analytical results linking Hadamard derivatives, Le Roy functions, and fractional hyper-Bessel equations, expanding understanding of these mathematical structures.

Findings

Connections between special functions and integro-differential equations established

New analytical results on Hadamard-type derivatives and Le Roy functions

Generalization of fractional hyper-Bessel equations with applications in probability

Abstract

In this paper we show several connections between special functions arising from generalized COM-Poisson-type statistical distributions and integro-differential equations with varying coefficients involving Hadamard-type operators. New analytical results are obtained, showing the particular role of Hadamard-type derivatives in connection with a recently introduced generalization of the Le Roy function. We are also able to prove a general connection between fractional hyper-Bessel-type equations involving Hadamard operators and Le Roy functions.