{
  "id": "0912.0917",
  "version": "v1",
  "published": "2009-12-04T19:04:19.000Z",
  "updated": "2009-12-04T19:04:19.000Z",
  "title": "A Note on the 2F1 Hypergeometric Function",
  "authors": [
    "Armen Bagdasaryan"
  ],
  "comment": "7 pages; accepted by J. Math. Res",
  "journal": "J. Math. Res. vol. 2, no. 3, (2010), 71--77",
  "categories": [
    "math.CA",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The special case of the hypergeometric function $_{2}F_{1}$ represents the binomial series $(1+x)^{\\alpha}=\\sum_{n=0}^{\\infty}(\\:\\alpha n\\:)x^{n}$ that always converges when $|x|<1$. Convergence of the series at the endpoints, $x=\\pm 1$, depends on the values of $\\alpha$ and needs to be checked in every concrete case. In this note, using new approach, we reprove the convergence of the hypergeometric series $_{2}F_{1}(\\alpha,\\beta;\\beta;x)$ for $|x|<1$ and obtain new result on its convergence at point $x=-1$ for every integer $\\alpha\\neq 0$. The proof is within a new theoretical setting based on the new method for reorganizing the integers and on the regular method for summation of divergent series.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-12-04T19:04:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "33C05"
    ],
    "keywords": [
      "2f1 hypergeometric function",
      "convergence",
      "binomial series",
      "divergent series",
      "hypergeometric series"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.0917B"
    }
  }
}