{
  "id": "math/0601303",
  "version": "v3",
  "published": "2006-01-12T22:27:02.000Z",
  "updated": "2007-01-23T13:06:58.000Z",
  "title": "The structure relation for Askey-Wilson polynomials",
  "authors": [
    "Tom H. Koornwinder"
  ],
  "comment": "18 pages, minor corrections and updated references",
  "journal": "J. Comput. Appl. Math. (2007), 214-226",
  "doi": "10.1016/j.cam.2006.10.015",
  "categories": [
    "math.CA"
  ],
  "abstract": "An explicit structure relation for Askey-Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey-Wilson inner product and which sends polynomials of degree n to polynomials of degree n+1. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtained for other families in the q-Askey scheme and the Askey scheme. This is explicitly discussed for Jacobi polynomials, continuous q-Jacobi polynomials, continuous q-ultraspherical polynomials, and for big q-Jacobi polynomials. An already known structure relation for this last family can be obtained from the new structure relation by using the three-term recurence relation and the second order q-difference formula. The results are also put in the framework of a more general theory. Their relationship with earlier work by Zhedanov and Bangerezako is discussed. There is also a connection with the string equation in discrete matrix models and with the Sklyanin algebra.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2007-01-23T13:06:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33D45",
      "33C45"
    ],
    "keywords": [
      "askey-wilson polynomials",
      "second order q-difference formula",
      "explicit structure relation",
      "discrete matrix models",
      "askey-wilson inner product"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Computational and Applied Mathematics",
      "year": 2007,
      "month": "Oct",
      "volume": 207,
      "number": 2,
      "pages": 214
    },
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007JCoAM.207..214K"
    }
  }
}