{"paper":{"title":"Factors of sums involving $q$-binomial coefficients and powers of $q$-integers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Su-Dan Wang, Victor J. W. Guo","submitted_at":"2017-01-24T03:36:08Z","abstract_excerpt":"We show that, for all positive integers $n_1, \\ldots, n_m$, $n_{m+1}=n_1$, and any non-negative integers $j$ and $r$ with $j\\leqslant m$, the expression $$ \\frac{1}{[n_1]}{n_1+n_{m}\\brack n_1}^{-1} \\sum_{k=1}^{n_1}[2k][k]^{2r}q^{jk^2-(r+1)k}\\prod_{i=1}^{m} {n_i+n_{i+1}\\brack n_i+k} $$ is a Laurent polynomial in $q$ with integer cofficients, where $[n]=1+q+\\cdots+q^{n-1}$ and ${n\\brack k}=\\prod_{i=1}^k(1-q^{n-i+1})/(1-q^i)$. This gives a $q$-analogue of a divisibility result on the Catalan triangle obtained by the first author and Zeng, and also confirms a conjecture of the first author and Zen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.07016","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"}