{
  "id": "math/0410320",
  "version": "v1",
  "published": "2004-10-13T14:48:17.000Z",
  "updated": "2004-10-13T14:48:17.000Z",
  "title": "Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour",
  "authors": [
    "A. Martinez-Finkelshtein",
    "R. Orive"
  ],
  "comment": "37 pages, 10 figures",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "Classical Jacobi polynomials $P_{n}^{(\\alpha,\\beta)}$, with $\\alpha, \\beta>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters $\\alpha_n,\\beta_n$ depend on $n$ in such a way that $$ \\lim_{n\\to\\infty}\\frac{\\alpha_{n}}{n}=A, \\quad \\lim_{n\\to\\infty}\\frac{\\beta_{n}}{n}=B, $$ with $A,B \\in \\mathbb{R}$. We restrict our attention to the case where the limits $A,B$ are not both positive and take values outside of the triangle bounded by the straight lines A=0, B=0 and $A+B+2=0$. As a corollary, we show that in the limit the zeros distribute along certain curves that constitute trajectories of a quadratic differential. The non-hermitian orthogonality relations for Jacobi polynomials with varying parameters lie in the core of our approach; in the cases we consider, these relations hold on a single contour of the complex plane. The asymptotic analysis is performed using the Deift-Zhou steepest descent method based on the Riemann-Hilbert reformulation of Jacobi polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-13T14:48:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45"
    ],
    "keywords": [
      "jacobi polynomials orthogonal",
      "single contour",
      "riemann-hilbert analysis",
      "deift-zhou steepest descent method",
      "non-hermitian orthogonality relations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10320M"
    }
  }
}