Free self-decomposability and unimodality of the Fuss-Catalan distributions

TL;DR

This paper investigates the properties of Fuss-Catalan distributions, focusing on conditions for free self-decomposability and unimodality, and establishes precise criteria for these properties.

Contribution

It provides the first characterization of free self-decomposability of Fuss-Catalan distributions based on parameters p and r.

Findings

Fuss-Catalan distribution $u(p,r)$ is freely self-decomposable if and only if 1 d7d7 p=r d7d7 2.

Identifies conditions for free infinite divisibility, free regularity, and unimodality of Fuss-Catalan distributions.

Establishes the relationship between distribution parameters and their free probabilistic properties.

Abstract

We study properties of the Fuss-Catalan distributions $\mu(p,r)$, $p\geq1$, $0<r\leq p$: free infinite divisibility, free self-decomposability, free regularity and unimodality. We show that the Fuss-Catalan distribution $\mu(p,r)$ is freely self-decomposable if and only if $1 \leq p=r \leq 2$.