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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1808.04717","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-08-13T15:55:28Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"07df5d6c9c5a2c559336d124b612b0a448803da25e0d6ae4a79c71b59e46e412","abstract_canon_sha256":"11cc81ed4975e66fefa3137b7a8b6583dea7ac997089d1d09b25f415baba1d64"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:10:54.187672Z","signature_b64":"T3MO81yxfSey6NqosXM7twkdzNsMRjclWuOE4IZ6JlQz14e8/IdO119cHan2/owHk2X6h2qAfVL4uNT46tQoAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"164ec4189d9192b135500bc60972789cb9949cf1c2379081e4a1b378e61807aa","last_reissued_at":"2026-07-05T01:10:54.187204Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:10:54.187204Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $|q|<1$. 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