{
  "id": "math/0510278",
  "version": "v1",
  "published": "2005-10-13T15:12:31.000Z",
  "updated": "2005-10-13T15:12:31.000Z",
  "title": "Type II Hermite-Padé approximation to the exponential function",
  "authors": [
    "A. B. J. Kuijlaars",
    "H. Stahl",
    "W. Van Assche",
    "F. Wielonsky"
  ],
  "comment": "20 pages, 5 figures",
  "journal": "Journal of Computational and Applied Mathematics 207 (2007), 227--244",
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "We obtain strong and uniform asymptotics in every domain of the complex plane for the scaled polynomials $a (3nz)$, $b (3nz)$, and $c (3nz)$ where $a$, $b$, and $c$ are the type II Hermite-Pad\\'e approximants to the exponential function of respective degrees $2n+2$, $2n$ and $2n$, defined by $a (z)e^{-z}-b (z)=\\O (z^{3n+2})$ and $a (z)e^{z}-c (z)={\\O}(z^{3n+2})$ as $z\\to 0$. Our analysis relies on a characterization of these polynomials in terms of a $3\\times 3$ matrix Riemann-Hilbert problem which, as a consequence of the famous Mahler relations, corresponds by a simple transformation to a similar Riemann-Hilbert problem for type I Hermite-Pad\\'e approximants. Due to this relation, the study that was performed in previous work, based on the Deift-Zhou steepest descent method for Riemann-Hilbert problems, can be reused to establish our present results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-10-13T15:12:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exponential function",
      "deift-zhou steepest descent method",
      "approximation",
      "hermite-pade approximants",
      "similar riemann-hilbert problem"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Computational and Applied Mathematics",
      "year": 2007,
      "month": "Oct",
      "volume": 207,
      "number": 2,
      "pages": 227
    },
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007JCoAM.207..227K"
    }
  }
}