{"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"}