{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 0$ be an integer.\n\n1. For all $x, y \\in \\C$,\n$$\\binom{x + y}{n} = \\sum_{k=0}^{n} \\binom{x}{k} \\binom{y}{n - k},$$\nwhere $\\binom{x}{k} = \\frac{x(x-1)\\cdots(x-k+1)}{k!}$ is the [[binomial coefficient|0GBZM0MT0F]].\n2. Equivalently, for all $b, c \\in \\C$ with $c \\notin \\set{0, -1, -2, \\ldots}$,\n$${}_2F_1(-n, b; c; 1) = \\frac{(c - b)_n}{(c)_n},$$\nwhere ${}_2F_1$ is the terminating [[hypergeometric series|428M453CR2]] and $(x)_n$ is the [[Pochhammer symbol|YZHM3HSM0F]].\n\n## Remarks\n\nBoth sides of part 1 are polynomials in $x$ and $y$.\nFor natural numbers $x = m$ and $y = p$ the identity is [[Vandermonde's identity|01JADB9VW5]], proved by counting $n$-element subsets of a union of an $m$-set and a $p$-set.\nA polynomial in two variables that vanishes on $\\N \\times \\N$ is zero, so part 1 holds for all $x, y$.\nPart 2 is also the case $a = -n$ of [[Gauss's hypergeometric theorem|1FC0T0GCCX]], since $\\frac{\\Gamma(c) \\Gamma(c + n - b)}{\\Gamma(c + n) \\Gamma(c - b)} = \\frac{(c - b)_n}{(c)_n}$.\n\nExample: with $x = y = \\frac12$ and $n = 2$, the left side is $\\binom{1}{2} = 0$, and the right side is $\\binom{1/2}{2} + \\binom{1/2}{1}^2 + \\binom{1/2}{2} = -\\frac18 + \\frac14 - \\frac18 = 0$.\n\nThe name records that the identity was known to Zhu Shijie long before Vandermonde.\n\n## Explanations\n\n### L1 human\n\nVandermonde's identity counts the ways to choose $n$ people from two groups: choose $k$ from the first and $n - k$ from the second.\nThe Chu–Vandermonde identity says that the same formula stays true when the group sizes are replaced by arbitrary numbers, even fractions or complex numbers.\nThis works because both sides are polynomials that agree at all whole numbers.\n\n### L2 human\n\nIn hypergeometric language it is the summation of a terminating ${}_2F_1$ at $z = 1$.\nIt is the simplest case of a whole hierarchy of terminating summations, followed by the Pfaff–Saalschütz formula for ${}_3F_2$ and Dougall's formula.\nIts $q$-analogue is the [[q-Vandermonde identity|MBRSCQRGGH]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "428M453CR2"
    ],
    "kind": "statement",
    "msc": [
      "33C05",
      "05A19"
    ],
    "name": "Chu–Vandermonde identity",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Vandermonde%27s_identity"
      }
    ],
    "tag": "4YK46R1QDR",
    "uses": [
      "0GBZM0MT0F",
      "YZHM3HSM0F",
      "428M453CR2"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/4YK46R1QDR/",
  "links": [
    {
      "field": "about",
      "target": "428M453CR2"
    },
    {
      "field": "uses",
      "target": "0GBZM0MT0F"
    },
    {
      "field": "uses",
      "target": "YZHM3HSM0F"
    },
    {
      "field": "uses",
      "target": "428M453CR2"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/4YK46R1QDR.md",
  "msc": [
    "33C05",
    "05A19"
  ],
  "name": "Chu–Vandermonde identity",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4YK/4YK46R1QDR.md",
  "status": null,
  "tag": "4YK46R1QDR",
  "uri": "m2://4YK46R1QDR",
  "url": "https://tamnd.github.io/math2/wiki/chu-vandermonde-identity/",
  "writing": {
    "tag": "4YK46R1QDR",
    "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
      }
    ]
  }
}