{
  "id": "1711.03349",
  "version": "v1",
  "published": "2017-11-09T12:28:15.000Z",
  "updated": "2017-11-09T12:28:15.000Z",
  "title": "A Characterization of Askey-Wilson polynomials",
  "authors": [
    "Maurice Kenfack Nangho",
    "Kerstin Jordaan"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "We show that the only monic orthogonal polynomials $\\{P_n\\}_{n=0}^{\\infty}$ that satisfy $$\\pi(x)\\mathcal{D}_{q}^2P_{n}(x)=\\sum_{j=-2}^{2}a_{n,n+j}P_{n+j}(x),\\; x=\\cos\\theta,\\;~ a_{n,n-2}\\neq 0,~ n=2,3,\\dots,$$ where $\\pi(x)$ is a polynomial of degree at most $4$ and $\\mathcal{D}_{q}$ is the Askey-Wilson operator, are Askey-Wilson polynomials and their special or limiting cases. This completes and proves a conjecture by Ismail concerning a structure relation satisfied by Askey-Wilson polynomials. We use the structure relation to derive upper bounds for the smallest zero and lower bounds for the largest zero of Askey-Wilson polynomials and their special cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-09T12:28:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "askey-wilson polynomials",
      "characterization",
      "structure relation",
      "monic orthogonal polynomials",
      "lower bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}