{
  "abstract": "We study the monic recurrence coefficients of the generalized Charlier polynomials associated with the discrete weight \\[ ρ_μ(k) = \\frac{μ^k}{(k!)^2}, \\qquad k\\in\\mathbb N_0, \\qquad μ>0. \\] We establish complete factorial asymptotics for the deviations $β_n-n$ and $μ-γ_n$, proving that they are beyond all algebraic orders and appear at the scale $μ^{n}/(n!)^2$. The key step is a parametrization $β_n-n=\\sqrtμ\\,c_nc_{n+1}$, $γ_n=μ(1-c_n^2)$, which reduces the problem to a positive sequence $(c_n)$ satisfying a discrete Painlevé II equation. Linearization around $c_n=0$ yields a discrete Bessel equation whose recessive solution is $J_n(2\\sqrtμ)$. We prove that the ratio $c_n/J_n(2\\sqrtμ)$ converges to a finite limit $L(μ)$, which we identify exactly as $L(μ)=1$ by combining Bessel comparison with a global analysis based on the sector decomposition of the associated Hankel determinant. This yields the exact connection formula \\[ c_n \\sim J_n(2\\sqrtμ), \\qquad n\\to\\infty, \\] and consequently the precise asymptotics \\[ β_n-n\\sim \\frac{μ^{n+1}}{n!(n+1)!}, \\qquad μ-γ_n\\sim \\frac{μ^{n+1}}{(n!)^2}. \\] We further develop a complementary approach based on Hankel determinants interpreted as a discrete logarithmic gas. We show that the Hankel determinant factorizes as $Δ_n(μ)=μ^{n(n-1)/2}T_n(μ)$ and admits a sector decomposition $T_n(μ)=1+\\sum_{q=1}^{n}S_q(μ)$ with sector contributions independent of $n$. We establish $S_q(μ)\\sim e^μμ^q/(q!)^2$, providing an independent derivation of the factorial corrections. Together, the Bessel comparison and sector decomposition reveal a rich integrable structure linking discrete Painlevé dynamics, Toda deformations, and Hankel--partition representations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00396",
  "author_total": 1,
  "authors": [
    {
      "name": "Mahouton Norbert Hounkonnou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6schn9zynxbma.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00396",
  "fulltext_url": null,
  "id": "6schn9zynxbma",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00396",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7216022496",
  "pdf_url": "https://arxiv.org/pdf/2610.00396",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6schn9zynxbma.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.00396",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00396"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Complete Factorial Asymptotics and Integrable Structure of Generalized Charlier Recurrence Coefficients",
  "title_text": "Complete Factorial Asymptotics and Integrable Structure of Generalized Charlier Recurrence Coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/6schn9zynxbma",
  "venue": "arXiv",
  "versions": []
}