{
  "id": "1602.00044",
  "version": "v1",
  "published": "2016-01-30T00:26:42.000Z",
  "updated": "2016-01-30T00:26:42.000Z",
  "title": "Bounds for Extreme Zeros of Quasi-orthogonal Ultraspherical Polynomials",
  "authors": [
    "Kathy Driver",
    "Martin E. Muldoon"
  ],
  "comment": "10 pages, 1 figure",
  "categories": [
    "math.CA"
  ],
  "abstract": "We discuss and compare upper and lower bounds obtained by two different methods for the positive zero of the ultraspherical polynomial $C_{n}^{(\\lambda)}$ that is greater than $1$ when $-3/2 < \\lambda < -1/2.$ Our first approach uses mixed three term recurrence relations and interlacing of zeros while the second approach uses a method going back to Euler and Rayleigh and already applied to Bessel functions and Laguerre and $q$-Laguerre polynomials. We use the bounds obtained by the second method to simplify the proof of the interlacing of the zeros of $(1-x^2)C_{n}^{(\\lambda)}$ and $C_{n+1}^{(\\lambda)}$, for $-3/2 < \\lambda < \\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-30T00:26:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45"
    ],
    "keywords": [
      "quasi-orthogonal ultraspherical polynomials",
      "extreme zeros",
      "term recurrence relations",
      "second method",
      "first approach"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160200044D"
    }
  }
}