{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nA function $F(n, k)$ of two integer variables is a *hypergeometric term* if the ratios $F(n + 1, k)/F(n, k)$ and $F(n, k + 1)/F(n, k)$ are [[rational functions|0XDK9CGYKJ]] of $n$ and $k$.\nA *Wilf–Zeilberger pair* (WZ pair) is a pair of hypergeometric terms $F$ and $G$ such that\n\n1. $F(n + 1, k) - F(n, k) = G(n, k + 1) - G(n, k)$ for all integers $n, k$;\n2. $G(n, k) = R(n, k)\\,F(n, k)$ for a rational function $R$, called the *WZ certificate*.\n\n## Remarks\n\nConsequence: suppose that for each $n$ the sum $\\sum_k F(n, k)$ converges and $G(n, k) \\to 0$ as $k \\to \\pm\\infty$.\nThen summing clause 1 over $k$ gives a [[telescoping sum|0GXKM1R7E4]] on the right.\nSo $\\sum_k F(n + 1, k) = \\sum_k F(n, k)$, and the sum is constant in $n$.\n\nExample: let $F(n, k) = \\binom{n}{k}/2^n$ and $G(n, k) = -\\binom{n}{k - 1}/2^{n + 1}$, with [[binomial coefficients|0GBZM0MT0F]] that vanish outside $0 \\le k \\le n$.\nPascal's rule gives $F(n + 1, k) - F(n, k) = \\big(\\binom{n}{k - 1} - \\binom{n}{k}\\big)/2^{n + 1}$, which is $G(n, k + 1) - G(n, k)$.\nThe certificate is $R(n, k) = -k/(2(n - k + 1))$.\nSo $\\sum_k \\binom{n}{k}/2^n$ is constant, and its value $1$ at $n = 0$ proves $\\sum_k \\binom{n}{k} = 2^n$.\n\nNon-example: the same $F$ with $G = 0$ is not a WZ pair, since clause 1 would need $F(n + 1, k) = F(n, k)$, which fails for $n = 0$, $k = 0$: $F(1, 0) = 1/2$ but $F(0, 0) = 1$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "WC0QNABPZ4",
    "kind": "definition",
    "msc": [
      "33F10",
      "05A19",
      "68W30"
    ],
    "name": "Wilf–Zeilberger pair",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Wilf%E2%80%93Zeilberger_pair"
      }
    ],
    "tag": "PW313WB7HE",
    "uses": [
      "0XDK9CGYKJ",
      "0GBZM0MT0F",
      "0GXKM1R7E4"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/PW313WB7HE/",
  "links": [
    {
      "field": "defines",
      "target": "WC0QNABPZ4"
    },
    {
      "field": "uses",
      "target": "0XDK9CGYKJ"
    },
    {
      "field": "uses",
      "target": "0GBZM0MT0F"
    },
    {
      "field": "uses",
      "target": "0GXKM1R7E4"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/PW313WB7HE.md",
  "msc": [
    "33F10",
    "05A19",
    "68W30"
  ],
  "name": "Wilf–Zeilberger pair",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PW3/PW313WB7HE.md",
  "status": null,
  "tag": "PW313WB7HE",
  "uri": "m2://PW313WB7HE",
  "url": "https://tamnd.github.io/math2/wiki/wilf-zeilberger-pair-definition/",
  "writing": {
    "tag": "PW313WB7HE",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}