{
  "abstract": "We study the monic polynomials $\\Sob{n}$ orthogonal with respect to a Sobolev-type modification of the discrete Krawtchouk inner product in which a single point mass at an exterior point $α<0$, outside the support $\\{0,1,\\dots,N\\}$ of the binomial measure, acts through the $j$-th forward difference $\\dif^{j}$ (here $\\Nzero=\\{0,1,2,\\dots\\}$, $\\Z_{+}=\\{1,2,\\dots\\}$, $\\R_{+}=(0,\\infty)$). In the joint scaling $n,N\\to\\infty$, $n/N\\to r$ with $0<p<r<1$ (the \\emph{reciprocal-Gamma} regime of Dominici's classification), we prove the Mehler--Heine formula. The exterior mass survives the large-degree limit exactly through the linear factor $z-α$, with the explicit universal constant $C_{\\ast}$, and the limit is independent of both $λ>0$ and $j\\ge1$; the same conclusion holds in the Uvarov case $j=0$ (Corollary~\\ref{cor:uvarov}). The derivation is pointwise and self-contained: it uses only a rank-one connection formula (Theorem~\\ref{thm:connection}), the reproducing-kernel recursion, Dominici's pointwise limit and the classical three-term recurrence; the normalised Sobolev correction obeys an exact affine recursion at fixed $N$, controlled by a triangular contraction argument over two backward windows with an explicit boundary estimate, and the contraction ratio $\\varrho<1$ is exactly the condition $r>p$. We also discuss the scale of the theorem: in this normalisation the classical term is asymptotically negligible, and the independence of $λ$ concerns each fixed $λ>0$, not the continuity at $λ=0$. Multiprecision computations illustrate the convergence and the exact value $C_{\\ast}=2.8$ for $p=0.3$, $r=3/5$, $α=-2.5$, $j=2$, $λ=5$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25060",
  "author_total": 2,
  "authors": [
    {
      "name": "Anier Soria Lorente"
    },
    {
      "name": "Junior Michel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0rhefmb0mkbyx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25060",
  "fulltext_url": null,
  "id": "0rhefmb0mkbyx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25060",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7214122694",
  "pdf_url": "https://arxiv.org/pdf/2609.25060",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0rhefmb0mkbyx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25060",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25060"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A reciprocal-Gamma Mehler--Heine limit for Krawtchouk--Sobolev type orthogonal polynomials with an exterior mass point",
  "title_text": "A reciprocal-Gamma Mehler--Heine limit for Krawtchouk--Sobolev type orthogonal polynomials with an exterior mass point",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/0rhefmb0mkbyx",
  "venue": "arXiv",
  "versions": []
}