{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:OITSXNMAZJZOLZS56RJKTJOAU5","short_pith_number":"pith:OITSXNMA","schema_version":"1.0","canonical_sha256":"72272bb580ca72e5e65df452a9a5c0a748b185d102ac21961287ae393820c902","source":{"kind":"arxiv","id":"1211.7259","version":1},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1211.7259","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-30T14:00:59Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"dfc1d92571f89c457ed374b1b9e7aede6ce3f93bb01db4e7279f83f51f56f941","abstract_canon_sha256":"2b1f93205bf76c55afb1cc0e6e3358d3bc6f38769da31bbaff922e20498cd0c6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:13.870960Z","signature_b64":"R+TKf8xqYCIeYO1fu4m01AmbboM9kxXx3o9quunjlWbzYLWxTs7OVqXB+EdNLbaCwO7MbHKo+83TT7AJMioeDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72272bb580ca72e5e65df452a9a5c0a748b185d102ac21961287ae393820c902","last_reissued_at":"2026-05-18T00:42:13.870453Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:13.870453Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1211.7259","created_at":"2026-05-18T00:42:13.870530+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.7259v1","created_at":"2026-05-18T00:42:13.870530+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.7259","created_at":"2026-05-18T00:42:13.870530+00:00"},{"alias_kind":"pith_short_12","alias_value":"OITSXNMAZJZO","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_16","alias_value":"OITSXNMAZJZOLZS5","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_8","alias_value":"OITSXNMA","created_at":"2026-05-18T12:27:16.716162+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OITSXNMAZJZOLZS56RJKTJOAU5","json":"https://pith.science/pith/OITSXNMAZJZOLZS56RJKTJOAU5.json","graph_json":"https://pith.science/api/pith-number/OITSXNMAZJZOLZS56RJKTJOAU5/graph.json","events_json":"https://pith.science/api/pith-number/OITSXNMAZJZOLZS56RJKTJOAU5/events.json","paper":"https://pith.science/paper/OITSXNMA"},"agent_actions":{"view_html":"https://pith.science/pith/OITSXNMAZJZOLZS56RJKTJOAU5","download_json":"https://pith.science/pith/OITSXNMAZJZOLZS56RJKTJOAU5.json","view_paper":"https://pith.science/paper/OITSXNMA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.7259&json=true","fetch_graph":"https://pith.science/api/pith-number/OITSXNMAZJZOLZS56RJKTJOAU5/graph.json","fetch_events":"https://pith.science/api/pith-number/OITSXNMAZJZOLZS56RJKTJOAU5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OITSXNMAZJZOLZS56RJKTJOAU5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OITSXNMAZJZOLZS56RJKTJOAU5/action/storage_attestation","attest_author":"https://pith.science/pith/OITSXNMAZJZOLZS56RJKTJOAU5/action/author_attestation","sign_citation":"https://pith.science/pith/OITSXNMAZJZOLZS56RJKTJOAU5/action/citation_signature","submit_replication":"https://pith.science/pith/OITSXNMAZJZOLZS56RJKTJOAU5/action/replication_record"}},"created_at":"2026-05-18T00:42:13.870530+00:00","updated_at":"2026-05-18T00:42:13.870530+00:00"}