{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:VNVXLRITGJY5ALPMDACB6YJJ75","short_pith_number":"pith:VNVXLRIT","schema_version":"1.0","canonical_sha256":"ab6b75c5133271d02dec18041f6129ff7211b39ada757f12ee0da4bdd90d0bd2","source":{"kind":"arxiv","id":"2505.22819","version":5},"attestation_state":"computed","paper":{"title":"A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Abdelhay Benmoussa","submitted_at":"2025-05-28T19:55:05Z","abstract_excerpt":"We establish a new identity linking Bernoulli, Stirling (first kind), and Bessel (first kind) numbers: \\[ \\sum_{k=0}^{n} 2^{\\,n-k}\\,s(n,k)\\,B_k \\;=\\; \\sum_{k=0}^{n} b(n,k)\\,\\frac{(-1)^k\\,k!}{k+1}. \\] This parallels the classical Stirling--Bernoulli relation \\[ B_n = \\sum_{k=0}^{n} S(n,k)\\,\\frac{(-1)^k\\,k!}{k+1}, \\] replacing $S(n,k)$ with $s(n,k)$ and $b(n,k)$, and thus revealing a new structural connection among these families of numbers."},"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":"2505.22819","kind":"arxiv","version":5},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.GM","submitted_at":"2025-05-28T19:55:05Z","cross_cats_sorted":[],"title_canon_sha256":"6b95e55fea143c0410ceb3f29112084ecc84346cd01ba7316ef71b5f51fc6af2","abstract_canon_sha256":"49f21dcd4d46a04ae378cd842b17e90cb946c0e5711a9188a4cc3f88d69e156c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T12:11:18.271686Z","signature_b64":"P1zdvnBW1yJssCeKkqnrLHgXvNPLTvmxpp0ZvUs/DyG4ctBahfIxcndn2lXw5vY6LesUBsivcT+yiWI+5gwqBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab6b75c5133271d02dec18041f6129ff7211b39ada757f12ee0da4bdd90d0bd2","last_reissued_at":"2026-07-05T12:11:18.271131Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T12:11:18.271131Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A New Identity Linking Bernoulli Numbers, Stirling Numbers of the First Kind, and Bessel Numbers of the First Kind","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Abdelhay Benmoussa","submitted_at":"2025-05-28T19:55:05Z","abstract_excerpt":"We establish a new identity linking Bernoulli, Stirling (first kind), and Bessel (first kind) numbers: \\[ \\sum_{k=0}^{n} 2^{\\,n-k}\\,s(n,k)\\,B_k \\;=\\; \\sum_{k=0}^{n} b(n,k)\\,\\frac{(-1)^k\\,k!}{k+1}. \\] This parallels the classical Stirling--Bernoulli relation \\[ B_n = \\sum_{k=0}^{n} S(n,k)\\,\\frac{(-1)^k\\,k!}{k+1}, \\] replacing $S(n,k)$ with $s(n,k)$ and $b(n,k)$, and thus revealing a new structural connection among these families of numbers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.22819","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.22819/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2505.22819","created_at":"2026-07-05T12:11:18.271194+00:00"},{"alias_kind":"arxiv_version","alias_value":"2505.22819v5","created_at":"2026-07-05T12:11:18.271194+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2505.22819","created_at":"2026-07-05T12:11:18.271194+00:00"},{"alias_kind":"pith_short_12","alias_value":"VNVXLRITGJY5","created_at":"2026-07-05T12:11:18.271194+00:00"},{"alias_kind":"pith_short_16","alias_value":"VNVXLRITGJY5ALPM","created_at":"2026-07-05T12:11:18.271194+00:00"},{"alias_kind":"pith_short_8","alias_value":"VNVXLRIT","created_at":"2026-07-05T12:11:18.271194+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/VNVXLRITGJY5ALPMDACB6YJJ75","json":"https://pith.science/pith/VNVXLRITGJY5ALPMDACB6YJJ75.json","graph_json":"https://pith.science/api/pith-number/VNVXLRITGJY5ALPMDACB6YJJ75/graph.json","events_json":"https://pith.science/api/pith-number/VNVXLRITGJY5ALPMDACB6YJJ75/events.json","paper":"https://pith.science/paper/VNVXLRIT"},"agent_actions":{"view_html":"https://pith.science/pith/VNVXLRITGJY5ALPMDACB6YJJ75","download_json":"https://pith.science/pith/VNVXLRITGJY5ALPMDACB6YJJ75.json","view_paper":"https://pith.science/paper/VNVXLRIT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2505.22819&json=true","fetch_graph":"https://pith.science/api/pith-number/VNVXLRITGJY5ALPMDACB6YJJ75/graph.json","fetch_events":"https://pith.science/api/pith-number/VNVXLRITGJY5ALPMDACB6YJJ75/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VNVXLRITGJY5ALPMDACB6YJJ75/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VNVXLRITGJY5ALPMDACB6YJJ75/action/storage_attestation","attest_author":"https://pith.science/pith/VNVXLRITGJY5ALPMDACB6YJJ75/action/author_attestation","sign_citation":"https://pith.science/pith/VNVXLRITGJY5ALPMDACB6YJJ75/action/citation_signature","submit_replication":"https://pith.science/pith/VNVXLRITGJY5ALPMDACB6YJJ75/action/replication_record"}},"created_at":"2026-07-05T12:11:18.271194+00:00","updated_at":"2026-07-05T12:11:18.271194+00:00"}