{"paper":{"title":"Factors of sums and alternating sums of products of $q$-binomial coefficients and powers of $q$-integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Su-Dan Wang, Victor J. W. Guo","submitted_at":"2017-05-08T11:13:06Z","abstract_excerpt":"We prove that, for all positive integers $n_1, \\ldots, n_m$, $n_{m+1}=n_1$, and non-negative integers $j$ and $r$ with $j\\leqslant m$, the following two expressions \\begin{align*} &\\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\\brack n_1}^{-1}\\sum_{k=0}^{n_1} q^{j(k^2+k)-(2r+1)k}[2k+1]^{2r+1}\\prod_{i=1}^m {n_i+n_{i+1}+1\\brack n_i-k},\\\\[5pt] &\\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\\brack n_1}^{-1}\\sum_{k=0}^{n_1}(-1)^k q^{{k\\choose 2}+j(k^2+k)-2rk}[2k+1]^{2r+1}\\prod_{i=1}^m {n_i+n_{i+1}+1\\brack n_i-k} \\end{align*} are Laurent polynomials in $q$ with integer coefficients, where $[n]=1+q+\\cdots+q^{n-1}$ and ${n\\brack k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.06236","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"}