{
  "abstract": "The para-Krawtchouk polynomials arose in the search for spin chains with perfect state transfer; the para-Racah, $q$-para-Racah and para-Bannai--Ito polynomials followed, obtained through singular truncations of families of the Askey scheme and used in turn to design spin chains. All are orthogonal on bi-lattices; the prefix goes back to the para-Krawtchouk case, whose spectrum is that of the parabose oscillator. The representations that these polynomials provide of the corresponding algebras --- Hahn, Racah, $q$-Racah, Bannai--Ito and complementary Bannai--Ito --- are shown to be reducible. Each decomposes into a direct sum of two irreducible modules of the same algebra, one attached to each of the two interlaced sub-lattices, and the parameters of the submodules are given. The para polynomials are thereby expressed in terms of the classical polynomials that the submodules carry, and the pairing $λ_n=λ_{N-n}$ of their eigenvalues is accounted for; the three features of the bi-lattice setting on which the reduction rests are identified. The properties of the para families are then presented in a unified way in the light of this decomposition, together with the constructions through which these families arise and the settings in which they have been put to use.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38211",
  "author_total": 6,
  "authors": [
    {
      "name": "Stéphane Z. Beaulac"
    },
    {
      "name": "Nicolas Crampé"
    },
    {
      "name": "Quentin Labriet"
    },
    {
      "name": "Lucia Morey"
    },
    {
      "name": "Rafael I. Nepomechie"
    },
    {
      "name": "Luc Vinet"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0esz2g2k8y343.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38211",
  "fulltext_url": null,
  "id": "0esz2g2k8y343",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38211",
  "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"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7215094356",
  "pdf_url": "https://arxiv.org/pdf/2609.38211",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0esz2g2k8y343.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:15Z",
      "native_id": "2609.38211",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38211"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Para families of orthogonal polynomials",
  "title_text": "Para families of orthogonal polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/0esz2g2k8y343",
  "venue": "arXiv",
  "versions": []
}