We prove that if $pgeq 1$ and $-1leq rleq p-1$ then the binomial sequence $binom{np+r}{n}$, $n=0,1,...$, is positive definite and is the moment sequence of a probability measure $ u(p,r)$, whose support is contained in $left[0,p^p(p-1)^{1-p}right]$. If $p>1$ is a rational number and $-1<rleq p-1$ then $ u(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 $ u(p,-1)$ and $ u(p,0)$ are certain free convolution powers of the Bernoulli distribution. Finally we prove that the binomial sequence $binom{np+r}{n}$ is positive definite if and only if either $pgeq 1$, $-1leq rleq p-1$ or $pleq 0$, $p-1leq r leq 0$. The measures corresponding to the latter case are reflections of the former ones.
Download