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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","authors_text":"Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-01-10T20:44:27Z","title":"On sums of Ap\\'ery polynomials and related congruences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.1946","kind":"arxiv","version":4},"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":"sha256:570ab515a6d2ba74041ea013c4768f3002a1af3cdbe2deeab7fafb9704f1eac9","target":"record","created_at":"2026-05-18T02:53:15Z","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":"c9030b2add50be5628978516f322e7a1f05fff4584c87b8a5bc5ed762eb17226","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-01-10T20:44:27Z","title_canon_sha256":"5ef06c555487c8c94491d4bcc2bf51dd581130bc9b3e2116be682c68abd7f080"},"schema_version":"1.0","source":{"id":"1101.1946","kind":"arxiv","version":4}},"canonical_sha256":"4352263b486c0ce152b67d0beb4d6bb135037b7085a6ec31022d3b95e9be6788","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4352263b486c0ce152b67d0beb4d6bb135037b7085a6ec31022d3b95e9be6788","first_computed_at":"2026-05-18T02:53:15.883891Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:53:15.883891Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dtrqlgWGD/yQh8N0E0xY54tG10YiPjOPkOv1z5YcZpJvNW2qppWWhUid0glb0yH2K7xYy73eQT9FL16VXw36Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:53:15.884528Z","signed_message":"canonical_sha256_bytes"},"source_id":"1101.1946","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:570ab515a6d2ba74041ea013c4768f3002a1af3cdbe2deeab7fafb9704f1eac9","sha256:f22976c50a74ebe4e3b047cb0d68d0ab29b8196da50aa9899bfed08d89bb59de"],"state_sha256":"d7ccd69115b1fdb7b21bf3e9caea27bb16e15528ac569540f58cb64c2b5844d3"}