{
  "abstract": "The ad-conditions of Duistermaat and Grünbaum characterize bispectrality for a pair of differential operators. M.~Reach observed that a sequence of functions which is simultaneously an eigenfunction of a second-order differential operator and satisfies a recurrence relation of arbitrary order must satisfy a different kind of ad-conditions. We prove that the Reach ad-conditions are also sufficient, by an argument which is purely algebraic. This motivated us to explore ad-conditions in two new directions: to Dunkl differential-difference operators, where the coefficients of the ad-condition become elements of the Dunkl algebra and the reflection enters as a coefficient, and to matrix-valued orthogonal polynomials, where we exhibit ad-conditions for several families of exceptional matrix-valued polynomials satisfying five- and seven-term recurrence relations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00219",
  "author_total": 3,
  "authors": [
    {
      "name": "I. Bono Parisi"
    },
    {
      "name": "F. A. Grünbaum"
    },
    {
      "name": "I. Zurrián"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/baras3ad177hv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00219",
  "fulltext_url": null,
  "id": "baras3ad177hv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00219",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00219",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/baras3ad177hv.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00219",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00219"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Ad-conditions for higher-order recurrences: Dunkl and matrix-valued settings",
  "title_text": "Ad-conditions for higher-order recurrences: Dunkl and matrix-valued settings",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/baras3ad177hv",
  "venue": "arXiv",
  "versions": []
}