{"paper":{"title":"Probability distributions with binomial moments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.PR","authors_text":"Karol A. Penson, Wojciech Mlotkowski","submitted_at":"2013-09-03T07:01:22Z","abstract_excerpt":"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 B"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0595","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}