{
  "id": "2002.02633",
  "version": "v1",
  "published": "2020-02-07T06:02:52.000Z",
  "updated": "2020-02-07T06:02:52.000Z",
  "title": "On the extreme zeros of Jacobi polynomials",
  "authors": [
    "Geno Nikolov"
  ],
  "comment": "12 pages, 1 figure",
  "categories": [
    "math.CA"
  ],
  "abstract": "By applying the Euler--Rayleigh methods to a specific representation of the Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, we derive upper and lower bound for $1-x_{nn}^2(\\lambda)$, with $x_{nn}(\\lambda)$ being the largest zero of the $n$-th ultraspherical polynomial $P_n^{(\\lambda)}$. For every fixed $\\lambda>-1/2$, the limit of the ratio of our upper and lower bounds for $1-x_{nn}^2(\\lambda)$ does not exceed $1.6$. This paper is a continuation of [1].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-07T06:02:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "42C05"
    ],
    "keywords": [
      "jacobi polynomials",
      "extreme zeros",
      "lower bound",
      "largest zero",
      "hypergeometric functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}