{
  "id": "1010.1420",
  "version": "v3",
  "published": "2010-10-07T12:57:23.000Z",
  "updated": "2013-12-29T23:22:02.000Z",
  "title": "On a continued fraction expansion for Euler's constant",
  "authors": [
    "Khodabakhsh Hessami Pilehrood",
    "Tatiana Hessami Pilehrood"
  ],
  "journal": "J. Number Theory 133 (2013) 769-786",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Recently, A. I. Aptekarev and his collaborators found a sequence of rational approximations to Euler's constant $\\gamma$ defined by a third-order homogeneous linear recurrence. In this paper, we give a new interpretation of Aptekarev's approximations in terms of Meijer $G$-functions and hypergeometric-type series. This approach allows us to describe a very general construction giving linear forms in 1 and $\\gamma$ with rational coefficients. Using this construction we find new rational approximations to $\\gamma$ generated by a second-order inhomogeneous linear recurrence with polynomial coefficients. This leads to a continued fraction (though not a simple continued fraction) for Euler's constant. It seems to be the first non-trivial continued fraction expansion convergent to Euler's constant sub-exponentially, the elements of which can be expressed as a general pattern. It is interesting to note that the same homogeneous recurrence generates a continued fraction for the Euler-Gompertz constant found by Stieltjes in 1895.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-12-29T23:22:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J70",
      "11B37",
      "11B65",
      "33C60",
      "33F10"
    ],
    "keywords": [
      "eulers constant",
      "rational approximations",
      "linear recurrence",
      "non-trivial continued fraction expansion convergent",
      "first non-trivial continued fraction expansion"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.1420H"
    }
  }
}