{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:INJCMO2INQGOCUVWPUF6WTLLWE","short_pith_number":"pith:INJCMO2I","schema_version":"1.0","canonical_sha256":"4352263b486c0ce152b67d0beb4d6bb135037b7085a6ec31022d3b95e9be6788","source":{"kind":"arxiv","id":"1101.1946","version":4},"attestation_state":"computed","paper":{"title":"On sums of Ap\\'ery polynomials and related congruences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2011-01-10T20:44:27Z","abstract_excerpt":"The Ap\\'ery polynomials are given by $$A_n(x)=\\sum_{k=0}^n\\binom nk^2\\binom{n+k}k^2x^k\\ \\ (n=0,1,2,\\ldots).$$ (Those $A_n=A_n(1)$ are Ap\\'ery numbers.) Let $p$ be an odd prime. We show that $$\\sum_{k=0}^{p-1}(-1)^kA_k(x)\\equiv\\sum_{k=0}^{p-1}\\frac{\\binom{2k}k^3}{16^k}x^k\\pmod{p^2},$$ and that $$\\sum_{k=0}^{p-1}A_k(x)\\equiv\\left(\\frac xp\\right)\\sum_{k=0}^{p-1}\\frac{\\binom{4k}{k,k,k,k}}{(256x)^k}\\pmod{p}$$ for any $p$-adic integer $x\\not\\equiv 0\\pmod p$. This enables us to determine explicitly $\\sum_{k=0}^{p-1}(\\pm1)^kA_k$ mod $p$, and $\\sum_{k=0}^{p-1}(-1)^kA_k$ mod $p^2$ in the case $p\\equiv 2"},"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":"1101.1946","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-01-10T20:44:27Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5ef06c555487c8c94491d4bcc2bf51dd581130bc9b3e2116be682c68abd7f080","abstract_canon_sha256":"c9030b2add50be5628978516f322e7a1f05fff4584c87b8a5bc5ed762eb17226"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:15.884528Z","signature_b64":"dtrqlgWGD/yQh8N0E0xY54tG10YiPjOPkOv1z5YcZpJvNW2qppWWhUid0glb0yH2K7xYy73eQT9FL16VXw36Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4352263b486c0ce152b67d0beb4d6bb135037b7085a6ec31022d3b95e9be6788","last_reissued_at":"2026-05-18T02:53:15.883891Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:15.883891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On sums of Ap\\'ery polynomials and related congruences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2011-01-10T20:44:27Z","abstract_excerpt":"The Ap\\'ery polynomials are given by $$A_n(x)=\\sum_{k=0}^n\\binom nk^2\\binom{n+k}k^2x^k\\ \\ (n=0,1,2,\\ldots).$$ (Those $A_n=A_n(1)$ are Ap\\'ery numbers.) Let $p$ be an odd prime. We show that $$\\sum_{k=0}^{p-1}(-1)^kA_k(x)\\equiv\\sum_{k=0}^{p-1}\\frac{\\binom{2k}k^3}{16^k}x^k\\pmod{p^2},$$ and that $$\\sum_{k=0}^{p-1}A_k(x)\\equiv\\left(\\frac xp\\right)\\sum_{k=0}^{p-1}\\frac{\\binom{4k}{k,k,k,k}}{(256x)^k}\\pmod{p}$$ for any $p$-adic integer $x\\not\\equiv 0\\pmod p$. This enables us to determine explicitly $\\sum_{k=0}^{p-1}(\\pm1)^kA_k$ mod $p$, and $\\sum_{k=0}^{p-1}(-1)^kA_k$ mod $p^2$ in the case $p\\equiv 2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1946","kind":"arxiv","version":4},"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":"1101.1946","created_at":"2026-05-18T02:53:15.884013+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.1946v4","created_at":"2026-05-18T02:53:15.884013+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.1946","created_at":"2026-05-18T02:53:15.884013+00:00"},{"alias_kind":"pith_short_12","alias_value":"INJCMO2INQGO","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_16","alias_value":"INJCMO2INQGOCUVW","created_at":"2026-05-18T12:26:32.869790+00:00"},{"alias_kind":"pith_short_8","alias_value":"INJCMO2I","created_at":"2026-05-18T12:26:32.869790+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/INJCMO2INQGOCUVWPUF6WTLLWE","json":"https://pith.science/pith/INJCMO2INQGOCUVWPUF6WTLLWE.json","graph_json":"https://pith.science/api/pith-number/INJCMO2INQGOCUVWPUF6WTLLWE/graph.json","events_json":"https://pith.science/api/pith-number/INJCMO2INQGOCUVWPUF6WTLLWE/events.json","paper":"https://pith.science/paper/INJCMO2I"},"agent_actions":{"view_html":"https://pith.science/pith/INJCMO2INQGOCUVWPUF6WTLLWE","download_json":"https://pith.science/pith/INJCMO2INQGOCUVWPUF6WTLLWE.json","view_paper":"https://pith.science/paper/INJCMO2I","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.1946&json=true","fetch_graph":"https://pith.science/api/pith-number/INJCMO2INQGOCUVWPUF6WTLLWE/graph.json","fetch_events":"https://pith.science/api/pith-number/INJCMO2INQGOCUVWPUF6WTLLWE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/INJCMO2INQGOCUVWPUF6WTLLWE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/INJCMO2INQGOCUVWPUF6WTLLWE/action/storage_attestation","attest_author":"https://pith.science/pith/INJCMO2INQGOCUVWPUF6WTLLWE/action/author_attestation","sign_citation":"https://pith.science/pith/INJCMO2INQGOCUVWPUF6WTLLWE/action/citation_signature","submit_replication":"https://pith.science/pith/INJCMO2INQGOCUVWPUF6WTLLWE/action/replication_record"}},"created_at":"2026-05-18T02:53:15.884013+00:00","updated_at":"2026-05-18T02:53:15.884013+00:00"}