{"paper":{"title":"Some q-analogues of supercongruences of Rodriguez-Villegas","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":"2014-01-23T14:06:26Z","abstract_excerpt":"We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}^2}{16^k} \\equiv (-1)^{\\frac{p-1}{2}}\\pmod{p^2} has the following two nice q-analogues with [p]=1+q+...+q^{p-1}: \\sum_{k=0}^{p-1}\\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{(1+\\varepsilon)k} &\\equiv (-1)^{\\frac{p-1}{2}}q^{\\frac{(p^2-1)\\varepsilon}{4}}\\pmod{[p]^2}, where (a;q)_0=1, (a;q)_n=(1-a)(1-aq)...(1-aq^{n-1}) for n=1,2,..., and \\varepsilon=\\pm1. Several related conjectures are also proposed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.5978","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"}