{"paper":{"title":"Factors of alternating sums of powers of $q$-Narayana numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Qiang-Qiang Jiang, Victor J. W. Guo","submitted_at":"2017-02-28T01:50:31Z","abstract_excerpt":"The $q$-Narayana numbers $N_q(n,k)$ and $q$-Catalan numbers $C_n(q)$ are respectively defined by $$ N_q(n,k)=\\frac{1-q}{1-q^n}{n\\brack k}{n\\brack k-1}\\quad\\text{and}\\quad C_n(q)=\\frac{1-q}{1-q^{n+1}}{2n\\brack n}, $$ where ${n\\brack k}=\\prod_{i=1}^{k}\\frac{1-q^{n-i+1}}{1-q^i}$. We prove that, for any positive integers $n$ and $r$, there holds \\begin{align*} \\sum_{k=-n}^{n}(-1)^{k}q^{jk^2+{k\\choose 2}}N_q(2n+1,n+k+1)^r \\equiv 0 \\pmod{C_n(q)}, \\end{align*} where $0\\leqslant j\\leqslant 2r-1$. We also propose several related conjectures."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00003","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"}