{
  "id": "math/0205175",
  "version": "v1",
  "published": "2002-05-15T18:08:02.000Z",
  "updated": "2002-05-15T18:08:02.000Z",
  "title": "Asymptotic zero behavior of Laguerre polynomials with negative parameter",
  "authors": [
    "A. B. J. Kuijlaars",
    "K. T-R McLaughlin"
  ],
  "comment": "28 pages, 10 figures",
  "journal": "Constructive. Approximation 20 (2004), 497-523",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "We consider Laguerre polynomials $L_n^{(\\alpha_n)}(nz)$ with varying negative parameters $\\alpha_n$, such that the limit $A = -\\lim_n \\alpha_n/n$ exists and belongs to $(0,1)$. For $A > 1$, it is known that the zeros accumulate along an open contour in the complex plane. For every $A \\in (0,1)$, we describe a one-parameter family of possible limit sets of the zeros. Under the condition that the limit $r= - \\lim_n \\frac{1}{n} \\log \\dist(\\alpha_n, \\mathbb Z)$ exists, we show that the zeros accumulate on $\\Gamma_r \\cup [\\beta_1,\\beta_2]$ with $\\beta_1$ and $\\beta_2$ only depending on $A$. For $r \\in [0,\\infty)$, $\\Gamma_r$ is a closed loop encircling the origin, which for $r = +\\infty$, reduces to the origin. This shows a great sensitivity of the zeros to $\\alpha_n$'s proximity to the integers. We use a Riemann-Hilbert formulation for the Laguerre polynomials, together with the steepest descent method of Deift and Zhou to obtain asymptotics for the polynomials, from which the zero behavior follows.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-05-15T18:08:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30E15",
      "33C45"
    ],
    "keywords": [
      "laguerre polynomials",
      "asymptotic zero behavior",
      "negative parameter",
      "zeros accumulate",
      "steepest descent method"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......5175K"
    }
  }
}