{
  "id": "1406.0969",
  "version": "v1",
  "published": "2014-06-04T08:27:32.000Z",
  "updated": "2014-06-04T08:27:32.000Z",
  "title": "Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions",
  "authors": [
    "Alfredo Deaño",
    "Arno B. J. Kuijlaars",
    "Pablo Román"
  ],
  "comment": "42 pages, 5 figures",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "We consider polynomials $P_n$ orthogonal with respect to the weight $J_{\\nu}$ on $[0,\\infty)$, where $J_{\\nu}$ is the Bessel function of order $\\nu$. Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros are complex and accumulate as $n \\to \\infty$ near the vertical line $\\textrm{Re}\\, z = \\frac{\\nu \\pi}{2}$. We prove this fact for the case $0 \\leq \\nu \\leq 1/2$ from strong asymptotic formulas that we derive for the polynomials $P_n$ in the complex plane. Our main tool is the Riemann-Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift-Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for $\\nu \\leq 1/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-04T08:27:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C47",
      "34M50",
      "30E15",
      "33C10"
    ],
    "keywords": [
      "bessel function",
      "zero distribution",
      "asymptotic behavior",
      "polynomials orthogonal",
      "deift-zhou steepest descent method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.0969D"
    }
  }
}