{
  "abstract": "The deformed \\(q\\)-exponential function \\[ e_q(z,u) = \\sum_{n=0}^{\\infty} u^{\\binom{n}{2}} \\frac{z^n}{(q;q)_n} \\] provides a common framework containing several classical \\(q\\)-exponential functions as particular cases, including the Jackson \\(q\\)-exponentials \\(e_q(z)\\) and \\(E_q(z)\\). In this paper we investigate the multiplicative inversion problem \\[ e_q(z,u)e_q(-z,v)=1, \\] and determine all pairs of deformation parameters \\((u,v)\\) for which this identity holds. To this end, we introduce a family of coefficient polynomials whose common zeros characterize the inversion property. A geometric analysis of the first nontrivial coefficients reduces the problem to two parameter branches. The symmetric branch is excluded through a parity phenomenon, while the nonsymmetric branch is completely determined by the first two coefficient constraints. As a consequence, we prove a rigidity theorem showing that \\[ e_q(z,u)e_q(-z,v)=1 \\] if and only if \\[ (u,v)=(1,q) \\qquad\\text{or}\\qquad (u,v)=(q,1). \\] Thus the classical Jackson inversion identity is rigid within the deformed family and no new multiplicative inversion identities arise from the deformation parameter.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14095",
  "author_total": 1,
  "authors": [
    {
      "name": "Ronald Orozco López"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7gw0m2rfjvps6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14095",
  "fulltext_url": null,
  "id": "7gw0m2rfjvps6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14095",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7213336887",
  "pdf_url": "https://arxiv.org/pdf/2609.14095",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7gw0m2rfjvps6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14095",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14095"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Rigidity Property of the Deformed q-Exponential Function",
  "title_text": "A Rigidity Property of the Deformed q-Exponential Function",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/7gw0m2rfjvps6",
  "venue": "arXiv",
  "versions": []
}