{
  "abstract": "We investigate orthogonality under the shift-invariant resolvent delta operator $\\mathscr L=D(I-D)^{-1}$. First, we prove a rigidity theorem: a monic orthogonal polynomial sequence is $\\mathscr L$-Appell if and only if it is a translated Laguerre sequence. We then move beyond the Appell setting and show that the diagonal family associated with $$\\mathcal S_\\beta=\\frac{\\beta}{\\beta+1}\\mathcal L^0+\\frac{1}{\\beta+1}\\delta$$ satisfies the exact Hahn relation $$\\mathscr L B_{n+1}(x;\\beta)=(n+1)B_n(x;\\beta+1).$$ A second rigidity result shows that a single pointwise relation $P_{n+1}(c)=\\theta\\,\\mathscr L P_{n+1}(c)$ already forces the orthogonality functional to be a translated Laguerre functional or a translated member of the diagonal family. Connection and structure relations, a semiclassical Pearson equation, a differential characterization, and a generating function complete the description. Thus the resolvent operator provides an explicit setting in which delta-operator lowering and semiclassical orthogonality are linked by exact structural identities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31805",
  "author_total": 2,
  "authors": [
    {
      "name": "Youssef Esstafa"
    },
    {
      "name": "Ridha Sfaxi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/533wtyk3kx4xc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31805",
  "fulltext_url": null,
  "id": "533wtyk3kx4xc",
  "issns": [],
  "landing_url": "https://hal.science/hal-05764115v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-ca"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7214567126",
  "pdf_url": "https://hal.science/hal-05764115/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/533wtyk3kx4xc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:41Z",
      "native_id": "hal-05764115",
      "source": "hal",
      "url": "https://hal.science/hal-05764115v1"
    },
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.31805",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31805"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hahn-Type Orthogonality for the Resolvent Delta Operator $\\mathscr L=D(I-D)^{-1}$",
  "title_text": "Hahn-Type Orthogonality for the Resolvent Delta Operator ℒ=D(I−D)^−1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/533wtyk3kx4xc",
  "venue": "HAL",
  "versions": []
}