{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:GJH3VAQ7CEKJ2K32Y6APOWK7OB","short_pith_number":"pith:GJH3VAQ7","schema_version":"1.0","canonical_sha256":"324fba821f11149d2b7ac780f7595f707a05e85e0f4eeb4479396c776bcaa7fc","source":{"kind":"arxiv","id":"1302.3738","version":1},"attestation_state":"computed","paper":{"title":"On the free Gamma distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.PR","authors_text":"Steen Thorbjornsen, Uffe Haagerup","submitted_at":"2013-02-15T12:43:00Z","abstract_excerpt":"For each positive number $\\alpha$ we study the analog $\\nu_alpha$ in free probability of the classical Gamma distribution with parameter $\\alpha$. We prove that $\\nu_\\alpha$ is absolutely continuous and establish the main properties of the density, including analyticity and unimodality. We study further the asymptotic behavior of $\\nu_\\alpha$ as $\\alpha\\downarrow0$."},"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":"1302.3738","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-02-15T12:43:00Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"6fcf168e3a9c01265a28d707d57175a9e0f94e9b833ff4b9d5e0167208864d4f","abstract_canon_sha256":"e2fc527d5bacca22e2f81d59bf14e0b6837752089362213f67765b47eee6ffc3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:35.931977Z","signature_b64":"kgkLqTRII0OqwiijP1ELfemHcRShhDe5VStVXCFaCynFbReKbWux+XbYqBxDKGABc5DkSBMKcUHmTzNbJNo+Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"324fba821f11149d2b7ac780f7595f707a05e85e0f4eeb4479396c776bcaa7fc","last_reissued_at":"2026-05-18T03:33:35.931214Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:35.931214Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the free Gamma distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.PR","authors_text":"Steen Thorbjornsen, Uffe Haagerup","submitted_at":"2013-02-15T12:43:00Z","abstract_excerpt":"For each positive number $\\alpha$ we study the analog $\\nu_alpha$ in free probability of the classical Gamma distribution with parameter $\\alpha$. We prove that $\\nu_\\alpha$ is absolutely continuous and establish the main properties of the density, including analyticity and unimodality. We study further the asymptotic behavior of $\\nu_\\alpha$ as $\\alpha\\downarrow0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.3738","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":"1302.3738","created_at":"2026-05-18T03:33:35.931329+00:00"},{"alias_kind":"arxiv_version","alias_value":"1302.3738v1","created_at":"2026-05-18T03:33:35.931329+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.3738","created_at":"2026-05-18T03:33:35.931329+00:00"},{"alias_kind":"pith_short_12","alias_value":"GJH3VAQ7CEKJ","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_16","alias_value":"GJH3VAQ7CEKJ2K32","created_at":"2026-05-18T12:27:45.050594+00:00"},{"alias_kind":"pith_short_8","alias_value":"GJH3VAQ7","created_at":"2026-05-18T12:27:45.050594+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/GJH3VAQ7CEKJ2K32Y6APOWK7OB","json":"https://pith.science/pith/GJH3VAQ7CEKJ2K32Y6APOWK7OB.json","graph_json":"https://pith.science/api/pith-number/GJH3VAQ7CEKJ2K32Y6APOWK7OB/graph.json","events_json":"https://pith.science/api/pith-number/GJH3VAQ7CEKJ2K32Y6APOWK7OB/events.json","paper":"https://pith.science/paper/GJH3VAQ7"},"agent_actions":{"view_html":"https://pith.science/pith/GJH3VAQ7CEKJ2K32Y6APOWK7OB","download_json":"https://pith.science/pith/GJH3VAQ7CEKJ2K32Y6APOWK7OB.json","view_paper":"https://pith.science/paper/GJH3VAQ7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1302.3738&json=true","fetch_graph":"https://pith.science/api/pith-number/GJH3VAQ7CEKJ2K32Y6APOWK7OB/graph.json","fetch_events":"https://pith.science/api/pith-number/GJH3VAQ7CEKJ2K32Y6APOWK7OB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GJH3VAQ7CEKJ2K32Y6APOWK7OB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GJH3VAQ7CEKJ2K32Y6APOWK7OB/action/storage_attestation","attest_author":"https://pith.science/pith/GJH3VAQ7CEKJ2K32Y6APOWK7OB/action/author_attestation","sign_citation":"https://pith.science/pith/GJH3VAQ7CEKJ2K32Y6APOWK7OB/action/citation_signature","submit_replication":"https://pith.science/pith/GJH3VAQ7CEKJ2K32Y6APOWK7OB/action/replication_record"}},"created_at":"2026-05-18T03:33:35.931329+00:00","updated_at":"2026-05-18T03:33:35.931329+00:00"}