{
  "id": "0712.0069",
  "version": "v1",
  "published": "2007-12-01T10:19:36.000Z",
  "updated": "2007-12-01T10:19:36.000Z",
  "title": "Generalized Bochner theorem: characterization of the Askey-Wilson polynomials",
  "authors": [
    "Luc Vinet",
    "Alexei Zhedanov"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "Assume that there is a set of monic polynomials $P_n(z)$ satisfying the second-order difference equation $$ A(s) P_n(z(s+1)) + B(s) P_n(z(s)) + C(s) P_n(z(s-1)) = \\lambda_n P_n(z(s)), n=0,1,2,..., N$$ where $z(s), A(s), B(s), C(s)$ are some functions of the discrete argument $s$ and $N$ may be either finite or infinite. The irreducibility condition $A(s-1)C(s) \\ne 0$ is assumed for all admissible values of $s$. In the finite case we assume that there are $N+1$ distinct grid points $z(s), \\: s=0,1,..., N$ such that $z(i) \\ne z(j), \\: i \\ne j$. If $N=\\infty$ we assume that the grid $z(s)$ has infinitely many different values for different values of $s$. In both finite and infinite cases we assume also that the problem is non-degenerate, i.e. $\\lambda_n \\ne \\lambda_m, n \\ne m$. Then we show that necessarily: (i) the grid $z(s)$ is at most quadratic or q-quadratic in $s$; (ii) corresponding polynomials $P_n(z)$ are at most the Askey-Wilson polynomials corresponding to the grid $z(s)$. This result can be considered as generalizing of the Bochner theorem (characterizing the ordinary classical polynomials) to generic case of arbitrary difference operator on arbitrary grids.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-12-01T10:19:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "42C05"
    ],
    "keywords": [
      "generalized bochner theorem",
      "askey-wilson polynomials",
      "characterization",
      "second-order difference equation",
      "distinct grid points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.0069V"
    }
  }
}