{"paper":{"title":"Densities of the Raney distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Karol A. Penson, Karol Zyczkowski, Wojciech Mlotkowski","submitted_at":"2012-11-30T14:00:59Z","abstract_excerpt":"We prove that if $p\\ge 1$ and $0< r\\le p$ then the sequence $\\binom{mp+r}{m}\\frac{r}{mp+r}$, $m=0,1,2,...$, is positive definite, more precisely, is the moment sequence of a probability measure $\\mu(p,r)$ with compact support contained in $[0,+\\infty)$. This family of measures encompasses the multiplicative free powers of the Marchenko-Pastur distribution as well as the Wigner's semicircle distribution centered at $x=2$. We show that if $p>1$ is a rational number, $0<r\\le p$, then $\\mu(p,r)$ is absolutely continuous and its density $W_{p,r}(x)$ can be expressed in terms of the Meijer and the g"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.7259","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"}