Densities of the Raney distributions

TL;DR

This paper proves the positivity and describes the density of a family of probability measures related to Raney distributions, which generalize classical distributions like Marchenko-Pastur and Wigner's semicircle, with explicit formulas for certain cases.

Contribution

It establishes the positive definiteness of a new family of sequences as moment sequences of probability measures and provides explicit density formulas involving special functions.

Findings

The sequence is a moment sequence of a probability measure with compact support.

Explicit density functions are derived for rational parameters, involving Meijer and hypergeometric functions.

Special cases yield elementary density functions, including free square and free square root of Marchenko-Pastur.

Abstract

We prove that if $p\ge 1$ and $0< r\le p$ then the sequence $\binom{mp+r}{m}\frac{r}{mp+r}$, $m=0,1,2,...$, is positive definite, more precisely, is the moment sequence of a probability measure $\mu(p,r)$ with compact support contained in $[0,+\infty)$. This family of measures encompasses the multiplicative free powers of the Marchenko-Pastur distribution as well as the Wigner's semicircle distribution centered at $x=2$. We show that if $p>1$ is a rational number, $0<r\le p$, then $\mu(p,r)$ is absolutely continuous and its density $W_{p,r}(x)$ can be expressed in terms of the Meijer and the generalized hypergeometric functions. In some cases, including the multiplicative free square and the multiplicative free square root of the Marchenko-Pastur measure, $W_{p,r}(x)$ turns out to be an elementary function.