{"paper":{"title":"On $q$-analogues of some series for $\\pi$ and $\\pi^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Christian Krattenthaler, Qing-Hu Hou, Zhi-Wei Sun","submitted_at":"2018-02-05T16:44:37Z","abstract_excerpt":"We obtain a new $q$-analogue of the classical Leibniz series $\\sum_{k=0}^\\infty(-1)^k/(2k+1)=\\pi/4$, namely \\begin{equation*} \\sum_{k=0}^\\infty\\frac{(-1)^kq^{k(k+3)/2}}{1-q^{2k+1}}=\\frac{(q^2;q^2)_{\\infty}(q^8;q^8)_{\\infty}}{(q;q^2)_{\\infty}(q^4;q^8)_{\\infty}}, \\end{equation*} where $q$ is a complex number with $|q|<1$. We also show that the Zeilberger-type series $\\sum_{k=1}^\\infty(3k-1)16^k/(k\\binom{2k}k)^3=\\pi^2/2$ has two $q$-analogues with $|q|<1$, one of which is $$\\sum_{n=0}^\\infty q^{n(n+1)/2} \\frac {1-q^{3n+2}} {1-q} \\cdot\\frac{(q;q)_n^3 (-q;q)_n}{(q^3;q^2)_{n}^3} = (1-q)^2 \\frac{(q^2"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01506","kind":"arxiv","version":5},"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"}