Probability distributions with binomial moments

Understand

We prove that if $p\geq 1$ and $-1\leq r\leq p-1$ then the binomial sequence $\binom{np+r}{n}$, $n=0,1,...$, is positive definite and is the moment sequence of a probability measure $\nu(p,r)$, whose support is contained in $\left[0,p^p(p-1)^{1-p}\right]$.

• If $p>1$ is a rational number and $-1<r\leq p-1$ then $\nu(p,r)$ is absolutely continuous and its density function $V_{p,r}$ can be expressed in terms of the Meijer $G$-function.
• In particular cases $V_{p,r}$ is an elementary function.
• We show that for $p>1$ the measures $\nu(p,-1)$ and $\nu(p,0)$ are certain free convolution powers of the Bernoulli distribution.