{
  "id": "1403.4014",
  "version": "v2",
  "published": "2014-03-17T07:14:20.000Z",
  "updated": "2014-03-23T18:24:30.000Z",
  "title": "Umbral \"classical\" polynomials",
  "authors": [
    "Alexei Zhedanov"
  ],
  "comment": "24 pages",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the umbral \"classical\" orthogonal polynomials with respect to a generalized derivative operator $\\cal D$ which acts on monomials as ${\\cal D} x^n = \\mu_n x^{n-1}$ with some coefficients $\\mu_n$. Let $P_n(x)$ be a set of orthogonal polynomials. Define the new polynomials $Q_n(x) =\\mu_{n+1}^{-1}{\\cal D} P_{n+1}(x)$. We find necessary and sufficient conditions when the polynomials $Q_n(x)$ will also be orthogonal. Apart from well known examples of the classical orthogonal polynomials we present a new example of umbral classical polynomials expressed in terms of elliptic functions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-23T18:24:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sufficient conditions",
      "classical orthogonal polynomials",
      "elliptic functions",
      "coefficients",
      "generalized derivative operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.4014Z"
    }
  }
}