{"paper":{"title":"A divisor generating q-series identity and its applications to probability theory and random graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.NT","authors_text":"Archit Agarwal, Bibekananda Maji, Pramod Eyyunni, Subhash Chand Bhoria","submitted_at":"2024-05-03T06:37:30Z","abstract_excerpt":"In I981, Uchimura studied a divisor generating $q$-series that has applications in probability theory and in the analysis of data structures, called heaps. Mainly, he proved the following identity. For $|q|<1$, \\begin{equation*} \\sum_{n=1}^\\infty n q^n (q^{n+1})_\\infty =\\sum_{n=1}^{\\infty} \\frac{(-1)^{n-1} q^{\\frac{n(n+1)}{2} } }{(1-q^n) ( q)_n } = \\sum_{n=1}^{\\infty} \\frac{ q^n }{1-q^n}. \\end{equation*} Over the years, this identity has been generalized by many mathematicians in different directions. Uchimura himself in 1987, Dilcher (1995), Andrews-Crippa-Simon (1997), and recently Gupta-Kum"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2405.01877","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2405.01877/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}