{"paper":{"title":"$q$-Analogues of some series for powers of $\\pi$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Qing-Hu Hou, Zhi-Wei Sun","submitted_at":"2018-08-13T15:55:28Z","abstract_excerpt":"We obtain $q$-analogues of several series for powers of $\\pi$. For example, the identity $$\\sum_{k=0}^\\infty\\frac{(-1)^k}{(2k+1)^3}=\\frac{\\pi^3}{32}$$ has the following $q$-analogue: \\begin{equation*} \\sum_{k=0}^\\infty(-1)^k\\frac{q^{2k}(1+q^{2k+1})}{(1-q^{2k+1})^3}=\\frac{(q^2;q^4)_{\\infty}^2(q^4;q^4)_{\\infty}^6} {(q;q^2)_{\\infty}^4}, \\end{equation*} where $q$ is any complex number with $|q|<1$. We also give $q$-analogues of four new series for powers of $\\pi$ found by the second author."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.04717","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1808.04717/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}