{
  "id": "1801.10554",
  "version": "v1",
  "published": "2018-01-31T17:08:26.000Z",
  "updated": "2018-01-31T17:08:26.000Z",
  "title": "Structure relations of classical orthogonal polynomials of the quadratic and $q$-quadratic variable",
  "authors": [
    "Maurice Kenfack Nangho",
    "Kerstin Jordaan"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We characterize polynomials that satisfy a first structure relation of the form \\begin{equation}\\label{1} \\pi(x)\\mathbb{D}_{x}^2P_{n}(x)=\\sum_{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\\quad a_{n,n-2}\\neq 0,\\quad n=2,3,\\dots \\end{equation} where $\\pi(x)$ is a polynomial of degree at most $4$ and $\\mathbb{D}_{x}$ is the divided-difference operator \\[\\nonumber \\mathbb{D}_{x}\\,f(x(t))={f(x(t+{1\\over 2}))-f(x(t-{1\\over 2}))\\over x(t+{1\\over 2})-x(t-{1\\over 2})}.\\] We prove an equivalence between the existence of a structure relation of type (1) satisfied by a sequence of monic orthogonal polynomials $\\{P_n\\}_{n=0}^{\\infty}$, the orthogonality of the second derivatives $\\{\\mathbb{D}_{x}^2P_n\\}_{n= 2}^{\\infty}$ and a generalized Sturm-Liouville type equation and show that the only monic orthogonal polynomials that satisfy (1) are Wilson polynomials, Continuous Dual-Hahn polynomials, Askey-Wilson polynomials and their special or limiting cases as one or more parameters tends to $\\infty$. This work extends our previous work (cf. [17]) concerning a conjecture due to Ismail. We also derive a second structure relation for polynomials satisfying (1).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-31T17:08:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33D45",
      "33C45"
    ],
    "keywords": [
      "classical orthogonal polynomials",
      "monic orthogonal polynomials",
      "quadratic variable",
      "first structure relation",
      "generalized sturm-liouville type equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}