{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $a, b, c \\in \\C$ with $c \\notin \\set{0, -1, -2, \\ldots}$ and $\\operatorname{Re}(c - a - b) \u003e 0$, and let $\\Gamma$ be the [[gamma function|0H5SYBSED9]].\nThen the [[hypergeometric series|428M453CR2]] converges at $z = 1$, and\n$${}_2F_1(a, b; c; 1) = \\frac{\\Gamma(c) \\, \\Gamma(c - a - b)}{\\Gamma(c - a) \\, \\Gamma(c - b)}.$$\n\n## Remarks\n\nWhen moreover $\\operatorname{Re} c \u003e \\operatorname{Re} b \u003e 0$, put $z = 1$ in [[Euler's integral representation|812THX456R]]: the integral becomes the beta integral $B(b, c - a - b)$, and the gamma factors simplify to the right side.\nAbel's theorem on power series identifies the value of the series at $1$ with the limit of the function as $z \\to 1^-$.\nThe general case follows by analytic continuation in the parameters, or from a contiguous relation.\n\nExample: for $a = b = 1$, $c = 3$ the series is $\\sum_{n \\ge 0} \\frac{n! \\, n!}{(n+2)!/2 \\cdot n!} = \\sum_{n \\ge 0} \\frac{2}{(n+1)(n+2)} = 2$, and the right side is $\\frac{\\Gamma(3) \\Gamma(1)}{\\Gamma(2) \\Gamma(2)} = 2$.\n\nWith $a = -n$ for an integer $n \\ge 0$, the theorem gives the [[Chu–Vandermonde identity|4YK46R1QDR]].\n\nThe condition $\\operatorname{Re}(c - a - b) \u003e 0$ is needed: for $a = b = 1$, $c = 2$ one has $c - a - b = 0$, and the series $\\sum_n \\frac{1}{n+1}$ diverges.\n\n## Explanations\n\n### L1 human\n\nThe hypergeometric series is an infinite sum depending on three parameters.\nAt the edge point $z = 1$ the sum can be computed in closed form, as a ratio of gamma function values.\nThis one formula evaluates a huge number of special series.\n\n### L2 human\n\nThe theorem is the basic summation formula for ${}_2F_1$, and most closed-form evaluations of ${}_3F_2$ and higher series, such as those of Pfaff–Saalschütz and Dixon, are its relatives.\nIt also gives the connection coefficients between the solutions at $0$ and at $1$ of the hypergeometric equation, since these are values of ${}_2F_1$ at $1$.\nThe terminating case, Chu–Vandermonde, is a polynomial identity behind many binomial sums.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "428M453CR2",
      "463X5V31D9"
    ],
    "kind": "statement",
    "msc": [
      "33C05",
      "33B15"
    ],
    "name": "Gauss's hypergeometric theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Hypergeometric_function"
      }
    ],
    "tag": "1FC0T0GCCX",
    "uses": [
      "428M453CR2",
      "0H5SYBSED9"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/1FC0T0GCCX/",
  "links": [
    {
      "field": "about",
      "target": "428M453CR2"
    },
    {
      "field": "about",
      "target": "463X5V31D9"
    },
    {
      "field": "uses",
      "target": "428M453CR2"
    },
    {
      "field": "uses",
      "target": "0H5SYBSED9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/1FC0T0GCCX.md",
  "msc": [
    "33C05",
    "33B15"
  ],
  "name": "Gauss's hypergeometric theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1FC/1FC0T0GCCX.md",
  "status": null,
  "tag": "1FC0T0GCCX",
  "uri": "m2://1FC0T0GCCX",
  "url": "https://tamnd.github.io/math2/wiki/gausss-hypergeometric-theorem/",
  "writing": {
    "tag": "1FC0T0GCCX",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}