Fuss-Catalan numbers in noncommutative probability

Abstract

We prove that if $p,r\in R,p\ge 1$ and $0lerlep$ then the Fuss-Catalan sequence $\left(\genfrac{}{}{0ex}{}{mp+r}{m}\right)\frac{r}{mp+r}$ 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 $0le2rlep$ and $r+1lep$ then $\mu (p,r)$ is $⊞$-infinitely divisible. As a by-product, we show that the sequence $\frac{m^m}{m!}$ is positive definite and the corresponding probability measure is $⊠$-infinitely divisible.