{
  "id": "1401.6754",
  "version": "v1",
  "published": "2014-01-27T07:16:55.000Z",
  "updated": "2014-01-27T07:16:55.000Z",
  "title": "Abstract \"hypergeometric\" orthogonal polynomials",
  "authors": [
    "Alexei Zhedanov"
  ],
  "comment": "20 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "We find all polynomials solutions $P_n(x)$ of the abstract \"hypergeometric\" equation $L P_n(x) = \\lambda_n P_n(x)$, where $L$ is a linear operator sending any polynomial of degree $n$ to a polynomial of the same degree with the property that $L$ is two-diagonal in the monomial basis, i.e. $L x^n = \\lambda_n x^n + \\mu_n x^{n-1}$ with arbitrary nonzero coefficients $\\lambda_n, \\mu_n$ . Under obvious nondegenerate conditions, the polynomial eigensolutions $L P_n(x) = \\lambda_n P_n(x)$ are unique. The main result of the paper is a classification of all {\\it orthogonal} polynomials $P_n(x)$ of such type, i.e. $P_n(x)$ are assumed to be orthogonal with respect to a nondegenerate linear functional $\\sigma$. We show that the only solutions are: Jacobi, Laguerre (correspondingly little $q$-Jacobi and little $q$-Laguerre and other special and degenerate cases), Bessel and little -1 Jacobi polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-01-27T07:16:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45"
    ],
    "keywords": [
      "orthogonal polynomials",
      "hypergeometric",
      "nondegenerate linear functional",
      "arbitrary nonzero coefficients",
      "jacobi polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1401.6754Z"
    }
  }
}