{"paper":{"title":"Super congruences involving Bernoulli and Euler polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2014-07-02T16:24:45Z","abstract_excerpt":"Let $p>3$ be a prime, and let $a$ be a rational p-adic integer. Let $\\{B_n(x)\\}$ and $\\{E_n(x)\\}$ denote the Bernoulli polynomials and Euler polynomials, respectively. In this paper we show that $$\\sum_{k=0}^{p-1}\\binom ak\\binom{-1-a}k\\equiv (-1)^{\\langle a\\rangle_p}+ p^2t(t+1)E_{p-3}(-a)\\pmod{p^3}$$ and for $a\\not\\equiv -\\frac 12\\pmod p$, $$\\sum_{k=0}^{p-1}\\binom ak\\binom{-1-a}k\\frac 1{2k+1}\\equiv \\frac{1+2t}{1+2a} +p^2\\frac{t(t+1)}{1+2a}B_{p-2}(-a)\\pmod{p^3},$$ where $\\langle a\\rangle_p\\in\\{0,1,\\ldots,p-1\\}$ satisfying $a\\equiv \\langle a\\rangle_p\\pmod p$ and $t=(a-\\langle a\\rangle_p)/p$. Tak"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.0636","kind":"arxiv","version":6},"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"}