{"paper":{"title":"Proof of a conjecture involving Sun polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Guo-Shuai Mao, Hao Pan, Victor J. W. Guo","submitted_at":"2015-10-27T01:28:23Z","abstract_excerpt":"The Sun polynomials $g_n(x)$ are defined by \\begin{align*} g_n(x)=\\sum_{k=0}^n{n\\choose k}^2{2k\\choose k}x^k. \\end{align*} We prove that, for any positive integer $n$, there hold \\begin{align*} &\\frac{1}{n}\\sum_{k=0}^{n-1}(4k+3)g_k(x) \\in\\mathbb{Z}[x],\\quad\\text{and}\\\\ &\\sum_{k=0}^{n-1}(8k^2+12k+5)g_k(-1)\\equiv 0\\pmod{n}. \\end{align*} The first one confirms a recent conjecture of Z.-W. Sun, while the second one partially answers another conjecture of Z.-W. Sun. We give three different proofs of the former. One of them depends on the following congruence: $$ {m+n-2\\choose m-1}{n\\choose m}{2n\\ch"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.04005","kind":"arxiv","version":2},"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"}