{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:JJ3AF6G52GSWDIMJBJ4AMJMDC7","short_pith_number":"pith:JJ3AF6G5","schema_version":"1.0","canonical_sha256":"4a7602f8ddd1a561a1890a7806258317f94f03db78650974b71b40175d4f2497","source":{"kind":"arxiv","id":"1505.01472","version":1},"attestation_state":"computed","paper":{"title":"Directional convexity and characterizations of Beta and Gamma functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Janu sz Matkowski, Martin Himmel","submitted_at":"2015-04-22T17:42:29Z","abstract_excerpt":"The logarithmic convexity of restrictions of the Beta functions to rays parallel to the main diagonal and the functional equation \\[ \\phi\\left( x+1\\right) =\\frac{x\\left( x+k\\right) }{\\left( 2x+k+1\\right) \\left( 2x+k\\right) }\\phi\\left( x\\right) ,\\ \\ \\ \\ \\ \\ x>0, \\] for $k>0$ allow to get a characterizations of the Beta function. This fact and a notion of the beta-type function lead to a new characterization of the Gamma function."},"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":"1505.01472","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-04-22T17:42:29Z","cross_cats_sorted":[],"title_canon_sha256":"6b37e39cc58b6932f9302fdf3678a6cc4fe095d0c251358422edcf0d6fe5b088","abstract_canon_sha256":"a04bc65391200f806e537506f3eb2e79132687ffeb3c34df33e5517a6e141620"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:49.191179Z","signature_b64":"VgSmd0d0koqKzbgCT0fuB5HBfRID+kPul32q86Tu17enQot4/yZ6oLOeZZ/QhAFIzcy/HZ8y1q5hl8MiVa0OCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a7602f8ddd1a561a1890a7806258317f94f03db78650974b71b40175d4f2497","last_reissued_at":"2026-05-18T02:16:49.190381Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:49.190381Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Directional convexity and characterizations of Beta and Gamma functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Janu sz Matkowski, Martin Himmel","submitted_at":"2015-04-22T17:42:29Z","abstract_excerpt":"The logarithmic convexity of restrictions of the Beta functions to rays parallel to the main diagonal and the functional equation \\[ \\phi\\left( x+1\\right) =\\frac{x\\left( x+k\\right) }{\\left( 2x+k+1\\right) \\left( 2x+k\\right) }\\phi\\left( x\\right) ,\\ \\ \\ \\ \\ \\ x>0, \\] for $k>0$ allow to get a characterizations of the Beta function. This fact and a notion of the beta-type function lead to a new characterization of the Gamma function."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.01472","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":"1505.01472","created_at":"2026-05-18T02:16:49.190555+00:00"},{"alias_kind":"arxiv_version","alias_value":"1505.01472v1","created_at":"2026-05-18T02:16:49.190555+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.01472","created_at":"2026-05-18T02:16:49.190555+00:00"},{"alias_kind":"pith_short_12","alias_value":"JJ3AF6G52GSW","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_16","alias_value":"JJ3AF6G52GSWDIMJ","created_at":"2026-05-18T12:29:27.538025+00:00"},{"alias_kind":"pith_short_8","alias_value":"JJ3AF6G5","created_at":"2026-05-18T12:29:27.538025+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/JJ3AF6G52GSWDIMJBJ4AMJMDC7","json":"https://pith.science/pith/JJ3AF6G52GSWDIMJBJ4AMJMDC7.json","graph_json":"https://pith.science/api/pith-number/JJ3AF6G52GSWDIMJBJ4AMJMDC7/graph.json","events_json":"https://pith.science/api/pith-number/JJ3AF6G52GSWDIMJBJ4AMJMDC7/events.json","paper":"https://pith.science/paper/JJ3AF6G5"},"agent_actions":{"view_html":"https://pith.science/pith/JJ3AF6G52GSWDIMJBJ4AMJMDC7","download_json":"https://pith.science/pith/JJ3AF6G52GSWDIMJBJ4AMJMDC7.json","view_paper":"https://pith.science/paper/JJ3AF6G5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1505.01472&json=true","fetch_graph":"https://pith.science/api/pith-number/JJ3AF6G52GSWDIMJBJ4AMJMDC7/graph.json","fetch_events":"https://pith.science/api/pith-number/JJ3AF6G52GSWDIMJBJ4AMJMDC7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JJ3AF6G52GSWDIMJBJ4AMJMDC7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JJ3AF6G52GSWDIMJBJ4AMJMDC7/action/storage_attestation","attest_author":"https://pith.science/pith/JJ3AF6G52GSWDIMJBJ4AMJMDC7/action/author_attestation","sign_citation":"https://pith.science/pith/JJ3AF6G52GSWDIMJBJ4AMJMDC7/action/citation_signature","submit_replication":"https://pith.science/pith/JJ3AF6G52GSWDIMJBJ4AMJMDC7/action/replication_record"}},"created_at":"2026-05-18T02:16:49.190555+00:00","updated_at":"2026-05-18T02:16:49.190555+00:00"}