{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $0 \u003c q \u003c 1$, and let $(a; q)_\\infty = \\prod_{k=0}^\\infty (1 - a q^k)$ be the infinite [[q-Pochhammer symbol|Y1X8ZZQ7VM]].\nFor a real number $x$ that is not in $\\set{0, -1, -2, \\ldots}$, the *$q$-gamma function* is\n$$\\Gamma_q(x) = (1 - q)^{1 - x} \\frac{(q; q)_\\infty}{(q^x; q)_\\infty}.$$\n\n## Remarks\n\nThe products converge because $\\sum q^k \u003c \\infty$, and $(q^x; q)_\\infty \\ne 0$ because $q^{x+k} \\ne 1$ for all $k \\ge 0$.\n\nExample: $\\Gamma_q(1) = 1$, since the two products are equal.\nAlso $\\Gamma_q(2) = (1-q)^{-1} \\frac{(q;q)_\\infty}{(q^2;q)_\\infty} = \\frac{1 - q}{1 - q} = 1$ and $\\Gamma_q(3) = \\frac{(1-q)(1-q^2)}{(1-q)^2} = 1 + q$.\nIn general $\\Gamma_q(x + 1) = [x]_q\\, \\Gamma_q(x)$ with $[x]_q = \\frac{1 - q^x}{1 - q}$, so $\\Gamma_q(n + 1) = [1]_q [2]_q \\cdots [n]_q$.\n\nNon-example of a valid argument: for $x = -1$ the factor $1 - q^{x+1}$ of $(q^x; q)_\\infty$ is $0$, so $\\Gamma_q$ has a pole there.\n\nFor each fixed $x$, $\\Gamma_q(x) \\to \\Gamma(x)$ as $q \\to 1^-$, where $\\Gamma$ is the [[gamma function|0H5SYBSED9]].\nA related definition exists for $q \u003e 1$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "68P2VKSEP5",
    "kind": "definition",
    "msc": [
      "33D05"
    ],
    "name": "q-gamma function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Q-gamma_function"
      }
    ],
    "tag": "YBSCJY6BY2",
    "uses": [
      "Y1X8ZZQ7VM",
      "0H5SYBSED9"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/YBSCJY6BY2/",
  "links": [
    {
      "field": "defines",
      "target": "68P2VKSEP5"
    },
    {
      "field": "uses",
      "target": "Y1X8ZZQ7VM"
    },
    {
      "field": "uses",
      "target": "0H5SYBSED9"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YBSCJY6BY2.md",
  "msc": [
    "33D05"
  ],
  "name": "q-gamma function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YBS/YBSCJY6BY2.md",
  "status": null,
  "tag": "YBSCJY6BY2",
  "uri": "m2://YBSCJY6BY2",
  "url": "https://tamnd.github.io/math2/wiki/q-gamma-function-definition/",
  "writing": {
    "tag": "YBSCJY6BY2",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}