{
  "abstract": "We present a new classification of the class of all the hypergeometric orthogonal polynomial sequences. The classification uses properties of the coefficients $α_n$ and $β_n$ in the three-term recurrence relation satisfied by the orthogonal polynomial sequences. Such coefficients are rational functions of $n$ and are determined by a set of six parameters. The partial fraction decomposition of $α_n$ is a linear combination of five linearly independent functions of $n$ with coefficients $d_j$, for $0 \\le j \\le 4$, that are polynomials in our six parameters. For each value of a parameter $r$, associated with the eigenvalues, we classify the $α_n$ according to the set of coefficients $d_j$ that are nonzero. There are certain particular values of $r$ that must be considered separately. Our classification yields a collection of 53 disjoint classes and it is different from the Askey scheme in several aspects. It is similar to the classifications proposed recently by Koornwinder.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.18578",
  "author_total": 1,
  "authors": [
    {
      "name": "Luis Verde-Star"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a1atx8qs16g64.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.18578",
  "fulltext_url": null,
  "id": "a1atx8qs16g64",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.18578",
  "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",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7165143897",
  "pdf_url": "https://arxiv.org/pdf/2606.18578",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a1atx8qs16g64.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2606.18578",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.18578"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Classification of the hypergeometric orthogonal polynomials via their recurrence coefficients",
  "title_text": "Classification of the hypergeometric orthogonal polynomials via their recurrence coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/a1atx8qs16g64",
  "venue": "arXiv",
  "versions": []
}