{
  "id": "2307.06668",
  "version": "v1",
  "published": "2023-07-13T10:21:26.000Z",
  "updated": "2023-07-13T10:21:26.000Z",
  "title": "Charting the $q$-Askey scheme. III. Verde-Star scheme for $q=1$",
  "authors": [
    "Tom H. Koornwinder"
  ],
  "comment": "16 pages, 2 figures; dedicated to the memory of Jos\\'e Carlos Petronilho",
  "categories": [
    "math.CA"
  ],
  "abstract": "Following Verde-Star, Linear Algebra Appl. 627 (2021), we label families of orthogonal polynomials in the $q=1$ Askey scheme together with their hypergeometric representations by three sequences $x_k, h_k, g_k$ of polynomials in $k$, two of degree 1 and one of degree 2, satisfying certain constraints. Except for the Hermite polynomials, this gives rise to a precise classification and a very simple uniform parametrization of these families together with their limit transitions. This is displayed in a graphical scheme. We also discuss limits from the $q$-case to the case $q=1$, although this cannot be done in a uniform way.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-13T10:21:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45"
    ],
    "keywords": [
      "askey scheme",
      "verde-star scheme",
      "linear algebra appl",
      "simple uniform parametrization",
      "hypergeometric representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}