{
  "id": "0807.3939",
  "version": "v2",
  "published": "2008-07-24T18:04:11.000Z",
  "updated": "2009-01-21T16:15:47.000Z",
  "title": "An extended class of orthogonal polynomials defined by a Sturm-Liouville problem",
  "authors": [
    "David Gomez-Ullate",
    "Niky Kamran",
    "Robert Milson"
  ],
  "comment": "25 pages, some remarks and references added",
  "journal": "J. Math. Anal. Appl. 359 (2009) 352-367",
  "doi": "10.1016/j.jmaa.2009.05.052",
  "categories": [
    "math-ph",
    "math.CA",
    "math.MP"
  ],
  "abstract": "We present two infinite sequences of polynomial eigenfunctions of a Sturm-Liouville problem. As opposed to the classical orthogonal polynomial systems, these sequences start with a polynomial of degree one. We denote these polynomials as $X_1$-Jacobi and $X_1$-Laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the the compact interval $[-1,1]$ or the half-line $[0,\\infty)$, respectively, and they are a basis of the corresponding $L^2$ Hilbert spaces. Moreover, we prove a converse statement similar to Bochner's theorem for the classical orthogonal polynomial systems: if a self-adjoint second order operator has a complete set of polynomial eigenfunctions $\\{p_i\\}_{i=1}^\\infty$, then it must be either the $X_1$-Jacobi or the $X_1$-Laguerre Sturm-Liouville problem. A Rodrigues-type formula can be derived for both of the $X_1$ polynomial sequences.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-01-21T16:15:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "33C47"
    ],
    "keywords": [
      "orthogonal polynomials",
      "extended class",
      "classical orthogonal polynomial systems",
      "definite inner product",
      "self-adjoint second order operator"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0807.3939G"
    }
  }
}