{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:Z6H623CRC3ZY6XJ5X7B7IK7ML2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1e184a2aaf118961c6bf876e3d0d7f0985a49e9d42feedc64ade9b0bf3c222b1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-01-27T20:34:32Z","title_canon_sha256":"7bab9395863577e88f9f5cc777e5a47f36d6ace0f618dcd7d9f04cdf36e1f761"},"schema_version":"1.0","source":{"id":"1101.5386","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1101.5386","created_at":"2026-05-18T04:03:20Z"},{"alias_kind":"arxiv_version","alias_value":"1101.5386v5","created_at":"2026-05-18T04:03:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1101.5386","created_at":"2026-05-18T04:03:20Z"},{"alias_kind":"pith_short_12","alias_value":"Z6H623CRC3ZY","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_16","alias_value":"Z6H623CRC3ZY6XJ5","created_at":"2026-05-18T12:26:47Z"},{"alias_kind":"pith_short_8","alias_value":"Z6H623CR","created_at":"2026-05-18T12:26:47Z"}],"graph_snapshots":[{"event_id":"sha256:82dd93a292c2fa4dd95c08fa49ff2e46ca0c9134c791549fb240497f1ad1ffae","target":"graph","created_at":"2026-05-18T04:03:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Zhi-Hong Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-01-27T20:34:32Z","title":"Generalized Legendre polynomials and related congruences modulo $p^2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.5386","kind":"arxiv","version":5},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c0c360cfcf974712ecb07e71345719225ff413625caba79c723244224c550db8","target":"record","created_at":"2026-05-18T04:03:20Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1e184a2aaf118961c6bf876e3d0d7f0985a49e9d42feedc64ade9b0bf3c222b1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-01-27T20:34:32Z","title_canon_sha256":"7bab9395863577e88f9f5cc777e5a47f36d6ace0f618dcd7d9f04cdf36e1f761"},"schema_version":"1.0","source":{"id":"1101.5386","kind":"arxiv","version":5}},"canonical_sha256":"cf8fed6c5116f38f5d3dbfc3f42bec5ea673c7629aa98dd2a28a7e654182ee25","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cf8fed6c5116f38f5d3dbfc3f42bec5ea673c7629aa98dd2a28a7e654182ee25","first_computed_at":"2026-05-18T04:03:20.327891Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:03:20.327891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RqCGH+I6G94HRLPczBJtFPrHzTw10IrvvDzng4kuuuWXzg+CAT/6szjRQpklk7uhKNcyxBX14DfVn+qlRRk+Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:03:20.328371Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.5386","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c0c360cfcf974712ecb07e71345719225ff413625caba79c723244224c550db8","sha256:82dd93a292c2fa4dd95c08fa49ff2e46ca0c9134c791549fb240497f1ad1ffae"],"state_sha256":"897067324ed51636e0368a5740d81135ce23aadbf35967592650a3ce798e8dd0"}