Wojciech Mlotkowski

摘要

We prove that if $p,rin{R}$, $pge1$ and $0le rle p$ then the Fuss-Catalan sequence $inom{mp+r}mrac{r}{mp+r}$ 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 $oxplus$-infinitely divisible. As a by-product, we show that the sequence $rac{m^m}{m!}$ is positive definite and the corresponding probability measure is $oxtimes$-infinitely divisible.