{"paper":{"title":"Some congruences involving powers of Delannoy polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Victor J. W. Guo","submitted_at":"2014-12-10T00:36:18Z","abstract_excerpt":"The Delannoy polynomial $D_n(x)$ is defined by $$ D_n(x)=\\sum_{k=0}^{n}{n\\choose k}{n+k\\choose k}x^k. $$ We prove that, if $x$ is an integer and $p$ is a prime not dividing $x(x+1)$, then \\begin{align*} \\sum_{k=0}^{p-1}(2k+1)D_k(x)^3 &\\equiv p\\left(\\frac{-4x-3}{p}\\right) \\pmod{p^2}, \\\\ \\sum_{k=0}^{p-1}(2k+1)D_k(x)^4 &\\equiv p \\pmod{p^2}, \\\\ \\sum_{k=0}^{p-1}(-1)^k(2k+1)D_k(x)^3 &\\equiv p\\left(\\frac{4x+1}{p}\\right) \\pmod{p^2}, \\end{align*} where $\\big(\\frac{\\cdot}{p}\\big)$ denotes the Legendre symbol. The first two congruences confirm a conjecture of Z.-W. Sun [Sci. China 57 (2014), 1375--1400]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.7724","kind":"arxiv","version":1},"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"}