{"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"},"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"}