{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:UPSBMZEX4P27H6BHDJFZ2STQMZ","short_pith_number":"pith:UPSBMZEX","schema_version":"1.0","canonical_sha256":"a3e4166497e3f5f3f8271a4b9d4a706659ade6cd665ea2abbaf884150ff4f17a","source":{"kind":"arxiv","id":"1802.01506","version":5},"attestation_state":"computed","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"},"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":"1802.01506","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-02-05T16:44:37Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"4fc4ba649ee99f1bfac6ec9aa38189318c150005d66be3af2bdab65d93848a4a","abstract_canon_sha256":"3bb48657eab53c6c487875baac7f105bd4782028e068de87b81a203504dddab7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:54:03.108303Z","signature_b64":"GWt8D0D8sLBzZLvLPLoqEvx9Xm1CVooh0F54RqLUoC0SJMkIDMDsAcjhDSII5cB3r7i2FLgRhSwxgnrdrSlXDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3e4166497e3f5f3f8271a4b9d4a706659ade6cd665ea2abbaf884150ff4f17a","last_reissued_at":"2026-05-17T23:54:03.107762Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:54:03.107762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1802.01506","created_at":"2026-05-17T23:54:03.107867+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.01506v5","created_at":"2026-05-17T23:54:03.107867+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.01506","created_at":"2026-05-17T23:54:03.107867+00:00"},{"alias_kind":"pith_short_12","alias_value":"UPSBMZEX4P27","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_16","alias_value":"UPSBMZEX4P27H6BH","created_at":"2026-05-18T12:32:56.356000+00:00"},{"alias_kind":"pith_short_8","alias_value":"UPSBMZEX","created_at":"2026-05-18T12:32:56.356000+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/UPSBMZEX4P27H6BHDJFZ2STQMZ","json":"https://pith.science/pith/UPSBMZEX4P27H6BHDJFZ2STQMZ.json","graph_json":"https://pith.science/api/pith-number/UPSBMZEX4P27H6BHDJFZ2STQMZ/graph.json","events_json":"https://pith.science/api/pith-number/UPSBMZEX4P27H6BHDJFZ2STQMZ/events.json","paper":"https://pith.science/paper/UPSBMZEX"},"agent_actions":{"view_html":"https://pith.science/pith/UPSBMZEX4P27H6BHDJFZ2STQMZ","download_json":"https://pith.science/pith/UPSBMZEX4P27H6BHDJFZ2STQMZ.json","view_paper":"https://pith.science/paper/UPSBMZEX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.01506&json=true","fetch_graph":"https://pith.science/api/pith-number/UPSBMZEX4P27H6BHDJFZ2STQMZ/graph.json","fetch_events":"https://pith.science/api/pith-number/UPSBMZEX4P27H6BHDJFZ2STQMZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/UPSBMZEX4P27H6BHDJFZ2STQMZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/UPSBMZEX4P27H6BHDJFZ2STQMZ/action/storage_attestation","attest_author":"https://pith.science/pith/UPSBMZEX4P27H6BHDJFZ2STQMZ/action/author_attestation","sign_citation":"https://pith.science/pith/UPSBMZEX4P27H6BHDJFZ2STQMZ/action/citation_signature","submit_replication":"https://pith.science/pith/UPSBMZEX4P27H6BHDJFZ2STQMZ/action/replication_record"}},"created_at":"2026-05-17T23:54:03.107867+00:00","updated_at":"2026-05-17T23:54:03.107867+00:00"}