Year: 2013
Type: preprint
Abstract: 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 $ν(p,r)$, whose support is contained in $\left[0,p^p(p-1)^{1-p}\right]$. If $p>1$ is a rational number and $-11$ the measures $ν(p,-1)$ and $ν(p,0)$ 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 $p\geq 1$, $-1\leq r\leq p-1$ or $p\leq 0$, $p-1\leq r \leq 0$. The measures corresponding to the latter case are reflections of the former ones.
Source: arXiv (Cornell University)
Authors: Mlotkowski, Wojciech, Penson, Karol A.
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Agence Nationale de la Recherche
Award: Physique combinatoire
OpenAlex ID: W2166949998