1211.7259 (Wojciech Mlotkowski et al.)

Densities of the Raney distributions    [PDF]

Wojciech Mlotkowski, Karol A. Penson, Karol Zyczkowski

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, $0View original: http://arxiv.org/abs/1211.7259