{"paper":{"title":"Factors of Alternating Sums of Products of Binomial and q-Binomial Coefficients","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Frederic Jouhet, Jiang Zeng, Victor J. W. Guo","submitted_at":"2005-11-25T19:47:22Z","abstract_excerpt":"In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\\leq j\\leq m-1, {n_1+n_{m}\\brack n_1}^{-1}\\sum_{k=-n_1}^{n_1}(-1)^kq^{jk^2+{k\\choose 2}} \\prod_{i=1}^m {n_i+n_{i+1}\\brack n_i+k}\\in \\N[q], which generalizes a result of Calkin [Acta Arith. 86 (1998), 17--26]. Moreover, we show that for all positive integers n, r and j, {2n\\brack n}^{-1}{2j\\brack j} \\sum_{k=j}^n(-1)^{n-k}q^{A}\\frac{1-q^{2k+1}}{1-q^{n+k+1}} {2n\\brack n-k}{k+j\\brack k-j}^r\\in N[q], where A=(r-1){n\\c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0511635","kind":"arxiv","version":4},"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"}