{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:3L3UGORTBDDJGQ44OMQRUIPMJO","short_pith_number":"pith:3L3UGORT","schema_version":"1.0","canonical_sha256":"daf7433a3308c693439c73211a21ec4bbbcabb82cb16b97e07f1c1e555d9022c","source":{"kind":"arxiv","id":"1112.4979","version":1},"attestation_state":"computed","paper":{"title":"On the Modular Behaviour of the Infinite Product $(1-x)(1-xq)(1-xq^2)(1-xq^3)...$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.NT","authors_text":"Changgui Zhang","submitted_at":"2011-12-21T10:34:31Z","abstract_excerpt":"Let $q=e^{2\\pi i\\tau}$, $\\Im\\tau>0$, $x=e^{2\\pi i\\xi}\\in\\CC$ and $(x;q)_\\infty=\\prod_{n\\ge 0}(1-xq^n)$. Let $(q,x)\\mapsto(q^*,\\iota_q x)$ be the classical modular substitution given by $q^*=e^{-2\\pi i/\\tau}$ and $\\iota_q x=e^{2\\pi i\\xi/{\\tau}}$. The main goal of this Note is to study the \"modular behaviour\" of the infinite product $(x;q)_\\infty$, this means, to compare the function defined by $(x;q)_\\infty$ with that given by $(\\iota_q x;q^*)_\\infty$. Inspired by the work of Stieltjes on some semi-convergent series, we are led to a \"closed\" analytic formula for $(x;q)_\\infty$ by means of the d"},"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":"1112.4979","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-12-21T10:34:31Z","cross_cats_sorted":["math.QA"],"title_canon_sha256":"7d9b4a8ea650e4fca01fa16cdbc70d1566b1788b5843e0b88bc9905c8587411f","abstract_canon_sha256":"5d079d19ea20e16480e163c2251e30c9d492071ecf26ece408abc28fcde6cf96"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:05:56.633490Z","signature_b64":"4Lx6BCK+IDdcHF3L1bX4zeQZZ6gtHNhJ54OQizAqHjO9lH88gaQRs9F5bW+Yq92xbJ6s0T3DSUR+Oezrn2LOCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"daf7433a3308c693439c73211a21ec4bbbcabb82cb16b97e07f1c1e555d9022c","last_reissued_at":"2026-05-18T04:05:56.632735Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:05:56.632735Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Modular Behaviour of the Infinite Product $(1-x)(1-xq)(1-xq^2)(1-xq^3)...$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA"],"primary_cat":"math.NT","authors_text":"Changgui Zhang","submitted_at":"2011-12-21T10:34:31Z","abstract_excerpt":"Let $q=e^{2\\pi i\\tau}$, $\\Im\\tau>0$, $x=e^{2\\pi i\\xi}\\in\\CC$ and $(x;q)_\\infty=\\prod_{n\\ge 0}(1-xq^n)$. 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Inspired by the work of Stieltjes on some semi-convergent series, we are led to a \"closed\" analytic formula for $(x;q)_\\infty$ by means of the d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.4979","kind":"arxiv","version":1},"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":"1112.4979","created_at":"2026-05-18T04:05:56.632855+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.4979v1","created_at":"2026-05-18T04:05:56.632855+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.4979","created_at":"2026-05-18T04:05:56.632855+00:00"},{"alias_kind":"pith_short_12","alias_value":"3L3UGORTBDDJ","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_16","alias_value":"3L3UGORTBDDJGQ44","created_at":"2026-05-18T12:26:18.847500+00:00"},{"alias_kind":"pith_short_8","alias_value":"3L3UGORT","created_at":"2026-05-18T12:26:18.847500+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/3L3UGORTBDDJGQ44OMQRUIPMJO","json":"https://pith.science/pith/3L3UGORTBDDJGQ44OMQRUIPMJO.json","graph_json":"https://pith.science/api/pith-number/3L3UGORTBDDJGQ44OMQRUIPMJO/graph.json","events_json":"https://pith.science/api/pith-number/3L3UGORTBDDJGQ44OMQRUIPMJO/events.json","paper":"https://pith.science/paper/3L3UGORT"},"agent_actions":{"view_html":"https://pith.science/pith/3L3UGORTBDDJGQ44OMQRUIPMJO","download_json":"https://pith.science/pith/3L3UGORTBDDJGQ44OMQRUIPMJO.json","view_paper":"https://pith.science/paper/3L3UGORT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.4979&json=true","fetch_graph":"https://pith.science/api/pith-number/3L3UGORTBDDJGQ44OMQRUIPMJO/graph.json","fetch_events":"https://pith.science/api/pith-number/3L3UGORTBDDJGQ44OMQRUIPMJO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3L3UGORTBDDJGQ44OMQRUIPMJO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3L3UGORTBDDJGQ44OMQRUIPMJO/action/storage_attestation","attest_author":"https://pith.science/pith/3L3UGORTBDDJGQ44OMQRUIPMJO/action/author_attestation","sign_citation":"https://pith.science/pith/3L3UGORTBDDJGQ44OMQRUIPMJO/action/citation_signature","submit_replication":"https://pith.science/pith/3L3UGORTBDDJGQ44OMQRUIPMJO/action/replication_record"}},"created_at":"2026-05-18T04:05:56.632855+00:00","updated_at":"2026-05-18T04:05:56.632855+00:00"}