{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:Z6H623CRC3ZY6XJ5X7B7IK7ML2","short_pith_number":"pith:Z6H623CR","schema_version":"1.0","canonical_sha256":"cf8fed6c5116f38f5d3dbfc3f42bec5ea673c7629aa98dd2a28a7e654182ee25","source":{"kind":"arxiv","id":"1101.5386","version":5},"attestation_state":"computed","paper":{"title":"Generalized Legendre polynomials and related congruences modulo $p^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2011-01-27T20:34:32Z","abstract_excerpt":"For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial $P_n(a,x)=\\sum_{k=0}^n\\b ak\\b{-1-a}k(\\frac{1-x}2)^k$. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related to $P_{p-1}(a,x)$. For example, we show that $P_{p-1}(a,x)\\e (-1)^{<a>_p}P_{p-1}(a,-x)\\mod {p^2}$, where $<a>_p$ is the least nonnegative residue of $a$ modulo $p$. We also generalize some congruences of Zhi-Wei Sun, and determine $\\sum_{k=0}^{p-1}\\binom{2k}k\\binom{3k}k{54^{-k}}$ and $\\sum_{k=0}^{p-1}\\binom ak\\binom{b-a}k\\mod {p^2}$, where $[x]$ is the great"},"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":"1101.5386","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-01-27T20:34:32Z","cross_cats_sorted":[],"title_canon_sha256":"7bab9395863577e88f9f5cc777e5a47f36d6ace0f618dcd7d9f04cdf36e1f761","abstract_canon_sha256":"1e184a2aaf118961c6bf876e3d0d7f0985a49e9d42feedc64ade9b0bf3c222b1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:20.328371Z","signature_b64":"RqCGH+I6G94HRLPczBJtFPrHzTw10IrvvDzng4kuuuWXzg+CAT/6szjRQpklk7uhKNcyxBX14DfVn+qlRRk+Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cf8fed6c5116f38f5d3dbfc3f42bec5ea673c7629aa98dd2a28a7e654182ee25","last_reissued_at":"2026-05-18T04:03:20.327891Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:20.327891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Generalized Legendre polynomials and related congruences modulo $p^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2011-01-27T20:34:32Z","abstract_excerpt":"For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial $P_n(a,x)=\\sum_{k=0}^n\\b ak\\b{-1-a}k(\\frac{1-x}2)^k$. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related to $P_{p-1}(a,x)$. For example, we show that $P_{p-1}(a,x)\\e (-1)^{<a>_p}P_{p-1}(a,-x)\\mod {p^2}$, where $<a>_p$ is the least nonnegative residue of $a$ modulo $p$. We also generalize some congruences of Zhi-Wei Sun, and determine $\\sum_{k=0}^{p-1}\\binom{2k}k\\binom{3k}k{54^{-k}}$ and $\\sum_{k=0}^{p-1}\\binom ak\\binom{b-a}k\\mod {p^2}$, where $[x]$ is the great"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.5386","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":""},"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":"1101.5386","created_at":"2026-05-18T04:03:20.327947+00:00"},{"alias_kind":"arxiv_version","alias_value":"1101.5386v5","created_at":"2026-05-18T04:03:20.327947+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.5386","created_at":"2026-05-18T04:03:20.327947+00:00"},{"alias_kind":"pith_short_12","alias_value":"Z6H623CRC3ZY","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_16","alias_value":"Z6H623CRC3ZY6XJ5","created_at":"2026-05-18T12:26:47.523578+00:00"},{"alias_kind":"pith_short_8","alias_value":"Z6H623CR","created_at":"2026-05-18T12:26:47.523578+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/Z6H623CRC3ZY6XJ5X7B7IK7ML2","json":"https://pith.science/pith/Z6H623CRC3ZY6XJ5X7B7IK7ML2.json","graph_json":"https://pith.science/api/pith-number/Z6H623CRC3ZY6XJ5X7B7IK7ML2/graph.json","events_json":"https://pith.science/api/pith-number/Z6H623CRC3ZY6XJ5X7B7IK7ML2/events.json","paper":"https://pith.science/paper/Z6H623CR"},"agent_actions":{"view_html":"https://pith.science/pith/Z6H623CRC3ZY6XJ5X7B7IK7ML2","download_json":"https://pith.science/pith/Z6H623CRC3ZY6XJ5X7B7IK7ML2.json","view_paper":"https://pith.science/paper/Z6H623CR","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1101.5386&json=true","fetch_graph":"https://pith.science/api/pith-number/Z6H623CRC3ZY6XJ5X7B7IK7ML2/graph.json","fetch_events":"https://pith.science/api/pith-number/Z6H623CRC3ZY6XJ5X7B7IK7ML2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Z6H623CRC3ZY6XJ5X7B7IK7ML2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Z6H623CRC3ZY6XJ5X7B7IK7ML2/action/storage_attestation","attest_author":"https://pith.science/pith/Z6H623CRC3ZY6XJ5X7B7IK7ML2/action/author_attestation","sign_citation":"https://pith.science/pith/Z6H623CRC3ZY6XJ5X7B7IK7ML2/action/citation_signature","submit_replication":"https://pith.science/pith/Z6H623CRC3ZY6XJ5X7B7IK7ML2/action/replication_record"}},"created_at":"2026-05-18T04:03:20.327947+00:00","updated_at":"2026-05-18T04:03:20.327947+00:00"}