{
  "id": "2203.10526",
  "version": "v1",
  "published": "2022-03-20T11:25:05.000Z",
  "updated": "2022-03-20T11:25:05.000Z",
  "title": "Hankel Determinant and Orthogonal Polynomials for a Perturbed Gaussian Weight: from Finite $n$ to Large $n$ Asymptotics",
  "authors": [
    "Chao Min",
    "Yang Chen"
  ],
  "comment": "28 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the monic polynomials orthogonal with respect to a symmetric perturbed Gaussian weight $$ w(x;t):=\\mathrm{e}^{-x^2}\\left(1+t\\: x^2\\right)^\\lambda,\\qquad x\\in \\mathbb{R}, $$ where $t> 0,\\;\\lambda\\in \\mathbb{R}$. This weight is related to the single-user MIMO systems in information theory. It is shown that the recurrence coefficient $\\beta_n(t)$ is related to a particular Painlev\\'{e} V transcendent, and the sub-leading coefficient $\\mathrm{p}(n,t)$ satisfies the Jimbo-Miwa-Okamoto $\\sigma$-form of the Painlev\\'{e} V equation. Furthermore, we derive the second-order difference equations satisfied by $\\beta_n(t)$ and $\\mathrm{p}(n,t)$, respectively. This enables us to obtain the large $n$ full asymptotic expansions for $\\beta_n(t)$ and $\\mathrm{p}(n,t)$ with the aid of Dyson's Coulomb fluid approach. We also consider the Hankel determinant $D_n(t)$, generated by the perturbed Gaussian weight. It is found that $H_n(t)$, a quantity allied to the logarithmic derivative of $D_n(t)$, can be expressed in terms of $\\beta_n(t)$ and $\\mathrm{p}(n,t)$. Based on this result, we obtain the large $n$ asymptotic expansion of $H_n(t)$ and then that of the Hankel determinant $D_n(t)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-20T11:25:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C05",
      "33E17",
      "41A60"
    ],
    "keywords": [
      "hankel determinant",
      "orthogonal polynomials",
      "dysons coulomb fluid approach",
      "second-order difference equations",
      "monic polynomials orthogonal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}