{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:CGZN3JYLCHI6XSLSSLAWMYYRZJ","short_pith_number":"pith:CGZN3JYL","schema_version":"1.0","canonical_sha256":"11b2dda70b11d1ebc97292c1666311ca6b51b8054e5d8ce0e7fd82141bb2ad46","source":{"kind":"arxiv","id":"1512.06753","version":1},"attestation_state":"computed","paper":{"title":"An extension of Van Vleck's functional equation for the sine","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Bouikhalene Belaid, Elqorachi Elhoucien","submitted_at":"2015-12-09T09:29:53Z","abstract_excerpt":"In \\cite{St3} H. Stetk\\ae r obtained the solutions of Van Vleck's functional equation for the sine $$f(x\\tau(y)z_0)-f(xyz_0) =2f(x)f(y),\\; x,y\\in G,$$ where $G$ is a semigroup, $\\tau$ is an involution of $G$ and $z_0$ is a fixed element in the center of $G$. The purpose of this paper is to determine the complex-valued solutions of the following extension of Van Vleck's functional equation for the sine $$\\mu(y)f(x\\tau(y)z_0)-f(xyz_0) =2f(x)f(y), \\;x,y\\in G,$$ where $\\mu$ : $G\\longrightarrow \\mathbb{C}$ is a multiplicative function such that $\\mu(x\\tau(x))=1$ for all $x\\in G$. Furthermore, we ob"},"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":"1512.06753","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-12-09T09:29:53Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e9691d5a90e823e11c0ee6548763ac6a3de48ef6a16fbc9735110e0fc6ad5344","abstract_canon_sha256":"cbe99aafd4a5136de4110251f268da4759c321e44acd38d020ebb2730c2eca3d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:23:59.761556Z","signature_b64":"c4u1ctvRozADF/0xdLed8QTtfi5xtJNpHRDnN5jOpcwSJuN5qWclo1UJAHnGlkQ0l/1Jz1c9u7pH/udYZE2xCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11b2dda70b11d1ebc97292c1666311ca6b51b8054e5d8ce0e7fd82141bb2ad46","last_reissued_at":"2026-05-18T01:23:59.760899Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:23:59.760899Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An extension of Van Vleck's functional equation for the sine","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Bouikhalene Belaid, Elqorachi Elhoucien","submitted_at":"2015-12-09T09:29:53Z","abstract_excerpt":"In \\cite{St3} H. Stetk\\ae r obtained the solutions of Van Vleck's functional equation for the sine $$f(x\\tau(y)z_0)-f(xyz_0) =2f(x)f(y),\\; x,y\\in G,$$ where $G$ is a semigroup, $\\tau$ is an involution of $G$ and $z_0$ is a fixed element in the center of $G$. The purpose of this paper is to determine the complex-valued solutions of the following extension of Van Vleck's functional equation for the sine $$\\mu(y)f(x\\tau(y)z_0)-f(xyz_0) =2f(x)f(y), \\;x,y\\in G,$$ where $\\mu$ : $G\\longrightarrow \\mathbb{C}$ is a multiplicative function such that $\\mu(x\\tau(x))=1$ for all $x\\in G$. Furthermore, we ob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.06753","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":"1512.06753","created_at":"2026-05-18T01:23:59.761011+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.06753v1","created_at":"2026-05-18T01:23:59.761011+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.06753","created_at":"2026-05-18T01:23:59.761011+00:00"},{"alias_kind":"pith_short_12","alias_value":"CGZN3JYLCHI6","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_16","alias_value":"CGZN3JYLCHI6XSLS","created_at":"2026-05-18T12:29:14.074870+00:00"},{"alias_kind":"pith_short_8","alias_value":"CGZN3JYL","created_at":"2026-05-18T12:29:14.074870+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/CGZN3JYLCHI6XSLSSLAWMYYRZJ","json":"https://pith.science/pith/CGZN3JYLCHI6XSLSSLAWMYYRZJ.json","graph_json":"https://pith.science/api/pith-number/CGZN3JYLCHI6XSLSSLAWMYYRZJ/graph.json","events_json":"https://pith.science/api/pith-number/CGZN3JYLCHI6XSLSSLAWMYYRZJ/events.json","paper":"https://pith.science/paper/CGZN3JYL"},"agent_actions":{"view_html":"https://pith.science/pith/CGZN3JYLCHI6XSLSSLAWMYYRZJ","download_json":"https://pith.science/pith/CGZN3JYLCHI6XSLSSLAWMYYRZJ.json","view_paper":"https://pith.science/paper/CGZN3JYL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.06753&json=true","fetch_graph":"https://pith.science/api/pith-number/CGZN3JYLCHI6XSLSSLAWMYYRZJ/graph.json","fetch_events":"https://pith.science/api/pith-number/CGZN3JYLCHI6XSLSSLAWMYYRZJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CGZN3JYLCHI6XSLSSLAWMYYRZJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CGZN3JYLCHI6XSLSSLAWMYYRZJ/action/storage_attestation","attest_author":"https://pith.science/pith/CGZN3JYLCHI6XSLSSLAWMYYRZJ/action/author_attestation","sign_citation":"https://pith.science/pith/CGZN3JYLCHI6XSLSSLAWMYYRZJ/action/citation_signature","submit_replication":"https://pith.science/pith/CGZN3JYLCHI6XSLSSLAWMYYRZJ/action/replication_record"}},"created_at":"2026-05-18T01:23:59.761011+00:00","updated_at":"2026-05-18T01:23:59.761011+00:00"}