{
  "id": "1501.06107",
  "version": "v1",
  "published": "2015-01-25T02:35:42.000Z",
  "updated": "2015-01-25T02:35:42.000Z",
  "title": "Root geometry of polynomial sequences I: Type $(0,1)$",
  "authors": [
    "J. L. Gross",
    "T. Mansour",
    "T. W. Tucker",
    "D. G. L. Wang"
  ],
  "comment": "24 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "This paper is concerned with the distribution in the complex plane of the roots of a polynomial sequence $\\{W_n(x)\\}_{n\\ge0}$ given by a recursion $W_n(x)=aW_{n-1}(x)+(bx+c)W_{n-2}(x)$, with $W_0(x)=1$ and $W_1(x)=t(x-r)$, where $a>0$, $b>0$, and $c,t,r\\in\\mathbb{R}$. Our results include proof of the distinct-real-rootedness of every such polynomial $W_n(x)$, derivation of the best bound for the zero-set $\\{x\\mid W_n(x)=0\\ \\text{for some $n\\ge1$}\\}$, and determination of three precise limit points of this zero-set. Also, we give several applications from combinatorics and topological graph theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-25T02:35:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial sequence",
      "root geometry",
      "precise limit points",
      "complex plane",
      "best bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150106107G"
    }
  }
}