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- W4290646985 abstract "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 $nu(p,r)$, whose support is contained in $left[0,p^p(p-1)^{1-p}right]$. If $p>1$ is a rational number and $-1<rleq p-1$ 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. 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." @default.
- W4290646985 created "2022-08-08" @default.
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- W4290646985 date "2013-09-03" @default.
- W4290646985 modified "2023-10-17" @default.
- W4290646985 title "Probability distributions with binomial moments" @default.
- W4290646985 doi "https://doi.org/10.48550/arxiv.1309.0595" @default.
- W4290646985 hasPublicationYear "2013" @default.
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