{"paper":{"title":"Supercongruences involving dual sequences","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":"2015-12-02T16:57:59Z","abstract_excerpt":"In this paper we study some sophisticated supercongruences involving dual sequences. For $n=0,1,2,\\ldots$ define $$d_n(x)=\\sum_{k=0}^n\\binom nk\\binom xk2^k$$ and $$s_n(x)=\\sum_{k=0}^n\\binom nk\\binom xk\\binom{x+k}k=\\sum_{k=0}^n\\binom nk(-1)^k\\binom xk\\binom{-1-x}k.$$ For any odd prime $p$ and $p$-adic integer $x$, we determine $\\sum_{k=0}^{p-1}(\\pm1)^kd_k(x)^2$ and $\\sum_{k=0}^{p-1}(2k+1)d_k(x)^2$ modulo $p^2$; for example, we establish the new $p$-adic congruence $$\\sum_{k=0}^{p-1}(-1)^kd_k(x)^2\\equiv(-1)^{\\langle x\\rangle_p}\\pmod{p^2},$$ where $\\langle x\\rangle_p$ denotes the least nonnegativ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.00712","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"}