{
  "id": "1907.08552",
  "version": "v1",
  "published": "2019-07-19T15:40:42.000Z",
  "updated": "2019-07-19T15:40:42.000Z",
  "title": "Roots of generalised Hermite polynomials when both parameters are large",
  "authors": [
    "Davide Masoero",
    "Pieter Roffelsen"
  ],
  "comment": "40 pages, 18 figures",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the roots of the generalised Hermite polynomials $H_{m,n}$ when both $m$ and $n$ are large. We prove that the roots, when appropriately rescaled, densely fill a bounded quadrilateral region, called the elliptic region, and organise themselves on a deformed rectangular lattice, as was numerically observed by Clarkson. We describe the elliptic region and the deformed lattice in terms of elliptic integrals and their degeneration. Keywords: Generalised Hermite polynomials; roots asymptotics; Painleve IV; Boutroux Curves; Tritronquee solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-19T15:40:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34M55",
      "34M56",
      "33C45",
      "34M60",
      "65H04"
    ],
    "keywords": [
      "generalised hermite polynomials",
      "parameters",
      "elliptic region",
      "deformed rectangular lattice",
      "bounded quadrilateral region"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}