{
  "id": "2109.14999",
  "version": "v2",
  "published": "2021-09-30T10:53:45.000Z",
  "updated": "2022-04-08T07:25:01.000Z",
  "title": "Distances of roots of classical orthogonal polynomials",
  "authors": [
    "Michael Voit"
  ],
  "comment": "In the revised version some estimates were improved, and the comments on the existing literature were extended",
  "categories": [
    "math.CA",
    "math.PR"
  ],
  "abstract": "Let $(P_N)_{N\\ge0}$ one of the classical sequences of orthogonal polynomials, i.e., Hermite, Laguerre or Jacobi polynomials. For the roots $z_{1,N},\\ldots, z_{N,N}$ of $P_N$ we derive lower estimates for $\\min_{i\\ne j}|z_{i,N}-z_{j,N}|$ and the distances from the boundary of the orthogonality intervals. The proofs are based on recent results on the eigenvalues of the covariance matrices in central limit theorems for associated $\\beta$-random matrix ensembles where these entities appear as entries, and where the eigenvalues of these matrices are known.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-04-08T07:25:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C45",
      "60B20",
      "60B10"
    ],
    "keywords": [
      "classical orthogonal polynomials",
      "central limit theorems",
      "random matrix ensembles",
      "eigenvalues",
      "covariance matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}