{
  "id": "2009.10196",
  "version": "v1",
  "published": "2020-09-21T22:08:56.000Z",
  "updated": "2020-09-21T22:08:56.000Z",
  "title": "Zeros of Jacobi and Ultraspherical polynomials",
  "authors": [
    "J. Arvesú",
    "K. Driver",
    "L. Littlejohn"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Suppose $\\{P_{n}^{(\\alpha, \\beta)}(x)\\}_{n=0}^\\infty $ is a sequence of Jacobi polynomials with $ \\alpha, \\beta >-1.$ We discuss special cases of a question raised by Alan Sokal at OPSFA in 2019, namely, whether the zeros of $ P_{n}^{(\\alpha,\\beta)}(x)$ and $ P_{n+k}^{(\\alpha + t, \\beta + s )}(x)$ are interlacing if $s,t >0$ and $ k \\in \\mathbb{N}.$ We consider two cases of this question for Jacobi polynomials of consecutive degree and prove that the zeros of $ P_{n}^{(\\alpha,\\beta)}(x)$ and $ P_{n+1}^{(\\alpha, \\beta + 1 )}(x),$ $ \\alpha > -1, \\beta > 0, $ $ n \\in \\mathbb{N},$ are partially, but in general not fully, interlacing depending on the values of $\\alpha, \\beta$ and $n.$ A similar result holds for the extent to which interlacing holds between the zeros of $ P_{n}^{(\\alpha,\\beta)}(x)$ and $ P_{n+1}^{(\\alpha + 1, \\beta + 1 )}(x),$ $ \\alpha >-1, \\beta > -1.$ It is known that the zeros of the equal degree Jacobi polynomials $ P_{n}^{(\\alpha,\\beta)}(x)$ and $ P_{n}^{(\\alpha - t, \\beta + s )}(x)$ are interlacing for $ \\alpha -t > -1, \\beta > -1, $ $0 \\leq t,s \\leq 2.$ We prove that partial, but in general not full, interlacing of zeros holds between the zeros of $ P_{n}^{(\\alpha,\\beta)}(x)$ and $ P_{n}^{(\\alpha + 1, \\beta + 1 )}(x),$ when $ \\alpha > -1, \\beta > -1.$ We provide numerical examples that confirm that the results we prove cannot be strengthened in general. The symmetric case $\\alpha = \\beta = \\lambda -1/2$ of the Jacobi polynomials is also considered. We prove that the zeros of the ultraspherical polynomials $ C_{n}^{(\\lambda)}(x)$ and $ C_{n + 1}^{(\\lambda +1)}(x),$ $ \\lambda > -1/2$ are partially, but in general not fully, interlacing. The interlacing of the zeros of the equal degree ultraspherical polynomials $ C_{n}^{(\\lambda)}(x)$ and $ C_{n}^{(\\lambda +3)}(x),$ $ \\lambda > -1/2,$ is also discussed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-21T22:08:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "interlacing",
      "equal degree jacobi polynomials",
      "similar result holds",
      "equal degree ultraspherical polynomials",
      "zeros holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}