{"paper":{"title":"$q$-Analogues of two Ramanujan-type formulas for $1/\\pi$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Ji-Cai Liu, Victor J. W. Guo","submitted_at":"2018-02-06T13:47:17Z","abstract_excerpt":"We give $q$-analogues of the following two Ramanujan-type formulas for $1/\\pi$: \\begin{align*} \\sum_{k=0}^\\infty (6k+1)\\frac{(\\frac{1}{2})_k^3}{k!^3 4^k} =\\frac{4}{\\pi} \\quad\\text{and}\\quad \\sum_{k=0}^\\infty (-1)^k(6k+1)\\frac{(\\frac{1}{2})_k^3}{k!^3 8^k } =\\frac{2\\sqrt{2}}{\\pi}. \\end{align*} Our proof is based on two $q$-WZ pairs found by the first author in his earlier work."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01944","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"}