{
  "id": "1106.3512",
  "version": "v2",
  "published": "2011-06-17T15:13:32.000Z",
  "updated": "2012-01-09T07:19:56.000Z",
  "title": "Dunkl shift operators and Bannai-Ito polynomials",
  "authors": [
    "Satoshi Tsujimoto",
    "Luc Vinet",
    "Alexei Zhedanov"
  ],
  "comment": "35 pages, to be published in Adv.Math",
  "categories": [
    "math.CA"
  ],
  "abstract": "We consider the most general Dunkl shift operator $L$ with the following properties: (i) $L$ is of first order in the shift operator and involves reflections; (ii) $L$ preserves the space of polynomials of a given degree; (iii) $L$ is potentially self-adjoint. We show that under these conditions, the operator $L$ has eigenfunctions which coincide with the Bannai-Ito polynomials. We construct a polynomial basis which is lower-triangular and two-diagonal with respect to the action of the operator $L$. This allows to express the BI polynomials explicitly. We also present an anti-commutator AW(3) algebra corresponding to this operator. From the representations of this algebra, we derive the structure and recurrence relations of the BI polynomials. We introduce new orthogonal polynomials - referred to as the complementary BI polynomials - as an alternative $q \\to -1$ limit of the Askey-Wilson polynomials. These complementary BI polynomials lead to a new explicit expression for the BI polynomials in terms of the ordinary Wilson polynomials.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-01-09T07:19:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "33C47",
      "42C05"
    ],
    "keywords": [
      "bannai-ito polynomials",
      "complementary bi polynomials",
      "general dunkl shift operator",
      "ordinary wilson polynomials",
      "orthogonal polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1106.3512T"
    }
  }
}