{
  "id": "2210.16605",
  "version": "v1",
  "published": "2022-10-29T14:05:10.000Z",
  "updated": "2022-10-29T14:05:10.000Z",
  "title": "On orthogonal polynomials with respect to a class of differential operators",
  "authors": [
    "Jorge A. Borrego-Morell"
  ],
  "journal": "Applied Mathematics and Computation 219 (2013) 7853-7871",
  "doi": "10.1016/j.amc.2013.01.069",
  "categories": [
    "math.CA"
  ],
  "abstract": "We consider orthogonal polynomials with respect to a linear differential operator $$\\mathcal{L}^{(M)}=\\sum_{k=0}^{M}\\rho_{k}(z)\\frac{d^k}{dz^k}, $$ where $\\{\\rho_k\\}_{k=0}^{M}$ are complex polynomials such that $deg[\\rho_k]\\leq k, 0\\leq k \\leq M$, with equality for at least one index. We analyze the uniqueness and zero location of these polynomials. An interesting phenomenon occurring in this kind of orthogonality is the existence of operators for which the associated sequence of orthogonal polynomials reduces to a finite set. For a given operator, we find a classification of the measures for which it is possible to guarantee the existence of an infinite sequence of orthogonal polynomials, in terms of a linear system of difference equations with varying coefficients. Also, for the case of a first-order differential operator, we locate the zeros and establish the strong asymptotic behavior of these polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-29T14:05:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "linear differential operator",
      "strong asymptotic behavior",
      "first-order differential operator",
      "orthogonal polynomials reduces",
      "complex polynomials"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}