{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $q \\in \\C$ with $q \\ne 0$ and $q \\ne 1$, let $n \\ge 1$, and let $U \\subseteq \\C$ be a set with $qU \\subseteq U$.\nLet $F$ be a given [[function|0EPAT51DXQ]] of $n + 2$ complex variables.\nA *$q$-difference equation of order $n$* for an unknown function $y \\colon U \\to \\C$ is a condition\n$$F\\big(x, y(x), y(qx), \\dots, y(q^n x)\\big) = 0 \\quad \\text{for all } x \\in U.$$\nIt is *linear* if it has the form\n$$a_n(x) y(q^n x) + \\cdots + a_1(x) y(qx) + a_0(x) y(x) = b(x)$$\nwith given functions $a_0, \\dots, a_n, b$ on $U$.\n\n## Remarks\n\nExample: let $0 \u003c \\abs{q} \u003c 1$ and $e_q(x) = \\sum_{k \\ge 0} x^k / [k]_q!$ for $\\abs{x} \u003c 1/\\abs{1 - q}$, where $[k]_q = (1 - q^k)/(1 - q)$ and $[k]_q! = [1]_q \\cdots [k]_q$.\nSince the [[q-derivative|PMJT3FZE0Y]] satisfies $D_q x^k = [k]_q x^{k-1}$, one gets $D_q e_q = e_q$.\nWritten out, this is the first-order linear $q$-difference equation $e_q(qx) = \\big(1 + (q - 1)x\\big) e_q(x)$.\n\nNon-example: $y(x + 1) = x\\, y(x)$, satisfied by the gamma function, relates $y$ at $x$ and at the translate $x + 1$.\nIt involves no dilate $q^j x$, so it is an ordinary difference equation and not a $q$-difference equation.\n\nThe $q$-derivative $D_q y(x) = \\big(y(qx) - y(x)\\big)/\\big((q - 1)x\\big)$ lets one rewrite any $q$-difference equation in terms of $D_q, D_q^2, \\dots$, as a $q$-analog of a differential equation.\nMany authors assume $\\abs{q} \\ne 1$, or at least that $q$ is not a root of unity.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "FQPYGXJJN0",
    "kind": "definition",
    "msc": [
      "39A13",
      "05A30"
    ],
    "name": "q-difference equation",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quantum_calculus"
      }
    ],
    "tag": "BWT2AAEK9R",
    "uses": [
      "PMJT3FZE0Y",
      "0EPAT51DXQ"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/BWT2AAEK9R/",
  "links": [
    {
      "field": "defines",
      "target": "FQPYGXJJN0"
    },
    {
      "field": "uses",
      "target": "PMJT3FZE0Y"
    },
    {
      "field": "uses",
      "target": "0EPAT51DXQ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/BWT2AAEK9R.md",
  "msc": [
    "39A13",
    "05A30"
  ],
  "name": "q-difference equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BWT/BWT2AAEK9R.md",
  "status": null,
  "tag": "BWT2AAEK9R",
  "uri": "m2://BWT2AAEK9R",
  "url": "https://tamnd.github.io/math2/wiki/q-difference-equation-definition/",
  "writing": {
    "tag": "BWT2AAEK9R",
    "kind": "definition",
    "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
      }
    ]
  }
}