{"paper":{"title":"Some congruences involving central q-binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Jiang Zeng, Victor J. W. Guo","submitted_at":"2009-10-19T14:23:19Z","abstract_excerpt":"Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$ \\sum_{k=0}^{n-1}(-1)^kq^{-{k+1\\choose 2}}{2k\\brack k}_q \\equiv (\\frac{n}{5}) q^{-\\lfloor n^4/5\\rfloor} \\pmod{\\Phi_n(q)}, $$ where $\\big(\\frac{n}{p}\\big)$ is the Legendre symbol and $\\Phi_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that $$ \\sum_{k=0}^{3^a m-1} q^{k}{2k\\brack k}_q &\\equiv 0 \\pmod{(1-q^{3^a})/(1-q)}, \\sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\\choose 2}}{2k\\brack k}_q &\\equiv 0 \\pmod{(1-q^{5^a})/(1-q)}, $$ for $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.3563","kind":"arxiv","version":3},"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"}