{
  "id": "1712.08039",
  "version": "v1",
  "published": "2017-12-21T16:06:00.000Z",
  "updated": "2017-12-21T16:06:00.000Z",
  "title": "Two asymptotic expansions for gamma function developed by Windschitl's formula",
  "authors": [
    "Zhen-Hang Yang",
    "Jing-Feng Tian"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CA"
  ],
  "abstract": "In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for $n\\in \\mathbb{N}$ with $n\\geq 4$, we have \\begin{equation*} \\Gamma \\left( x+1\\right) =\\sqrt{2\\pi x}\\left( \\tfrac{x}{e}\\right) ^{x}\\left( x\\sinh \\tfrac{1}{x}\\right) ^{x/2}\\exp \\left( \\sum_{k=3}^{n-1}\\tfrac{\\left( 2k\\left( 2k-2\\right) !-2^{2k-1}\\right) B_{2k}}{2k\\left( 2k\\right) !x^{2k-1}} +R_{n}\\left( x\\right) \\right) \\end{equation*} with \\begin{equation*} \\left| R_{n}\\left( x\\right) \\right| \\leq \\frac{\\left| B_{2n}\\right| }{2n\\left( 2n-1\\right) }\\frac{1}{x^{2n-1}} \\end{equation*} for all $x>0$, where $B_{2n}$ is the Bernoulli number. Moreover, we present some approximation formulas for gamma function related to Windschitl's approximation one, which have higher accuracy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-12-21T16:06:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33B15",
      "41A60",
      "41A10",
      "41A20"
    ],
    "keywords": [
      "gamma function",
      "asymptotic expansions",
      "windschitls formula",
      "windschitls approximation formula",
      "power series"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}