{"paper":{"title":"On series expansions of zeros of the deformed exponential function","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO","math.CV"],"primary_cat":"math.CA","authors_text":"Alexey Kuznetsov","submitted_at":"2024-12-03T14:04:50Z","abstract_excerpt":"For $q \\in (0, 1)$, the deformed exponential function $f(x) = \\sum_{n \\geq 1} x^n q^{n(n-1)/2}/n!$ is known to have infinitely many simple and negative zeros $\\{x_k(q)\\}_{k \\geq 1}$. In this paper, we analyze the series expansions of $-x_k(q)/k$ and $k/x_k(q)$ in powers of $q$. We prove that the coefficients of these expansions are rational functions of the form $P_n(k)/Q_n(k)$ and $\\widehat{P}_n(k)/Q_n(k)$, where $Q_n(k) \\in {\\mathbb Z}[k]$ is explicitly defined and the polynomials $P_n(k), \\widehat{P}_n(k)\\in {\\mathbb Z}[k]$ can be computed recursively. We provide explicit formulas for the l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.02462","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/2412.02462/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"}