Probability distributions with binomial moments

We prove that if $p\ge 1$ and $-1\le r\le p-1$ then the binomial sequence $\left(\genfrac{}{}{0px}{}{np+r}{n}\right)$, $n=0,1,\mathrm{.}\mathrm{.}\mathrm{.}$, is positive definite and is the moment sequence of a probability measure $\nu (p,r)$, whose support is contained in $[0,p^p(p-1{)}^{1-p}]$. If $p>1$ is a rational number and $-1<r\leq p-1$then$\nu(p,r)$isabsolutelycontinuousanditsdensityfunction$V_{p,r}$canbeexpressedintermsoftheMeijer$G$-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. Finally we prove that the binomial sequence $\left(\genfrac{}{}{0px}{}{np+r}{n}\right)$ is positive definite if and only if either $p\ge 1$, $-1\le r\le p-1$ or $p\leq 0$,$p-1\leq r \leq 0$. The measures corresponding to the latter case are reflections of the former ones.