{
  "id": "1610.07309",
  "version": "v1",
  "published": "2016-10-24T07:27:10.000Z",
  "updated": "2016-10-24T07:27:10.000Z",
  "title": "Zeros of orthogonal polynomials near an algebraic singularity of the measure",
  "authors": [
    "Árpád Baricz",
    "Tivadar Danka"
  ],
  "comment": "25 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper we study the local zero behavior of orthogonal polynomials around an algebraic singularity, that is, when the measure of orthogonality is supported on $ [-1,1] $ and behaves like $ h(x)|x - x_0|^\\lambda dx $ for some $ x_0 \\in (-1,1) $, where $ h(x) $ is strictly positive and analytic. We shall sharpen the theorem of Yoram Last and Barry Simon and show that the so-called fine zero spacing (which is known for $ \\lambda = 0$) unravels in the general case, and the asymptotic behavior of neighbouring zeros around the singularity can be described with the zeros of the function $ c J_{\\frac{\\lambda - 1}{2}}(x) + d J_{\\frac{\\lambda + 1}{2}}(x) $, where $ J_a(x) $ denotes the Bessel function of the first kind and order $ a $. Moreover, using Sturm-Liouville theory, we study the behavior of this linear combination of Bessel functions, thus providing estimates for the zeros in question.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-24T07:27:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C05",
      "33C10",
      "33C45"
    ],
    "keywords": [
      "orthogonal polynomials",
      "algebraic singularity",
      "bessel function",
      "local zero behavior",
      "linear combination"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}