Fuss-Catalan numbers in noncommutative probability

Abstract

We prove that if $p,r\in{R}, p\ge1$ and $0le rle p$ then the Fuss-Catalan sequence $\binom{mp+r}m\frac{r}{mp+r}$ is positive definite. We study the family of the corresponding probability measures $\mu(p,r)$ on ${R}$ from the point of view of noncommutative probability. For example, we prove that if $0le 2rle p$ and $r+1le p$ then $\mu(p,r)$ is $\boxplus$-infinitely divisible. As a by-product, we show that the sequence $\frac{m^m}{m!}$ is positive definite and the corresponding probability measure is $\boxtimes$-infinitely divisible.