{
  "id": "1905.02397",
  "version": "v1",
  "published": "2019-05-07T08:19:37.000Z",
  "updated": "2019-05-07T08:19:37.000Z",
  "title": "Gegenbauer and other planar orthogonal polynomials on an ellipse in the complex plane",
  "authors": [
    "G. Akemann",
    "T. Nagao",
    "I. Parra",
    "G. Vernizzi"
  ],
  "comment": "37 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We show that several families of classical orthogonal polynomials on the real line are also orthogonal on the interior of an ellipse in the complex plane, subject to a weighted planar Lebesgue measure. In particular these include Gegenbauer polynomials $C_n^{(1+\\alpha)}(z)$ for $\\alpha>-1$ containing the Legendre polynomials $P_n(z)$, and the subset $P_n^{(\\alpha+\\frac12,\\pm\\frac12)}(z)$ of the Jacobi polynomials. These polynomials provide an orthonormal basis and the corresponding weighted Bergman space forms a complete metric space. This leads to a certain family of Selberg integrals in the complex plane. We recover the known orthogonality of Chebyshev polynomials of first up to fourth kind. The limit $\\alpha\\to\\infty$ leads back to the known Hermite polynomials orthogonal in the entire complex plane. When the ellipse degenerates to a circle we obtain the weight function and monomials known from the determinantal point process of the ensemble of truncated unitary random matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-07T08:19:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex plane",
      "planar orthogonal polynomials",
      "gegenbauer",
      "hermite polynomials orthogonal",
      "corresponding weighted bergman space forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}