{
  "id": "0912.2185",
  "version": "v1",
  "published": "2009-12-11T10:18:18.000Z",
  "updated": "2009-12-11T10:18:18.000Z",
  "title": "A Pick function related to the sequence of volumes of the unit ball in n-space",
  "authors": [
    "Christian Berg",
    "Henrik L. Pedersen"
  ],
  "categories": [
    "math.CA",
    "math.CV"
  ],
  "abstract": "We show that F_a(x)=\\frac{\\ln \\Gamma (x+1)}{x\\ln(ax)} is a Pick function for a\\ge 1 and find its integral representation. We also consider the function f(x)=(\\frac{\\pi^{x/2}}{\\Gamma(1+x/2)})^{1/(x\\ln x)} and show that \\ln f(x+1) is a Stieltjes function and that f(x+1) is completely monotonic on (0,\\infty). In particular f(n)=\\Omega_n^{1/(n\\ln n)},n\\ge 2 is a Hausdorff moment sequence. Here \\Omega_n is the volume of the unit ball in Euclidean n-space",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-11T10:18:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33B15",
      "30E20",
      "30E15"
    ],
    "keywords": [
      "unit ball",
      "pick function",
      "hausdorff moment sequence",
      "euclidean n-space",
      "stieltjes function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.2185B"
    }
  }
}