{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $a, b, c, q, z \\in \\C$ with $0 \u003c \\abs{q} \u003c 1$, $\\abs{z} \u003c 1$, $0 \u003c \\abs{b} \u003c 1$, and with $c$ and $az$ not of the form $q^{-m}$ for an integer $m \\ge 0$.\nThen\n$${}_2\\phi_1 \\left( \\begin{matrix} a, b \\\\ c \\end{matrix} ; q, z \\right) = \\frac{(b; q)_\\infty\\, (az; q)_\\infty}{(c; q)_\\infty\\, (z; q)_\\infty}\\; {}_2\\phi_1 \\left( \\begin{matrix} c/b, z \\\\ az \\end{matrix} ; q, b \\right),$$\nwhere ${}_2\\phi_1$ is the [[basic hypergeometric series|EB4N9RQR51]] and $(\\cdot\\,; q)_\\infty$ is the [[q-Pochhammer symbol|Y1X8ZZQ7VM]].\n\n## Remarks\n\nProof idea: write $\\frac{(b; q)_n}{(c; q)_n} = \\frac{(b; q)_\\infty}{(c; q)_\\infty} \\cdot \\frac{(c q^n; q)_\\infty}{(b q^n; q)_\\infty}$.\nExpand $\\frac{(c q^n; q)_\\infty}{(b q^n; q)_\\infty}$ by the [[q-binomial theorem|VVT27RB46B]] as a series in $b q^n$, exchange the two sums, and sum over $n$ with the $q$-binomial theorem again.\nExample: for $c = b$ the left side is ${}_1\\phi_0(a; -; q, z) = \\frac{(az; q)_\\infty}{(z; q)_\\infty}$, and on the right ${}_2\\phi_1(1, z; az; q, b) = 1$ since $(1; q)_n = 0$ for $n \\ge 1$; both sides agree.\nIterating the transformation gives two more forms, and the three are often called Heine's first, second and third transformations.\nThe third form is ${}_2\\phi_1(a, b; c; q, z) = \\frac{(abz/c; q)_\\infty}{(z; q)_\\infty}\\, {}_2\\phi_1(c/a, c/b; c; q, abz/c)$, valid when both series converge.\n\n## Explanations\n\n### L1 human\n\nHeine's transformation rewrites one basic hypergeometric series as an infinite product times another basic hypergeometric series with permuted parameters.\nThe roles of the variable $z$ and the parameter $b$ are exchanged.\nSuch a change can turn a slowly converging series into a fast one, or a hard sum into an easy one.\n\n### L2 human\n\nIt is the first and most used transformation of the ${}_2\\phi_1$ series, and its third form is the $q$-analogue of Euler's transformation of the [[hypergeometric series|428M453CR2]] ${}_2F_1$.\nSetting $z = c/(ab)$ and summing the right side with the $q$-binomial theorem gives the $q$-Gauss sum ${}_2\\phi_1(a, b; c; q, c/(ab)) = \\frac{(c/a, c/b; q)_\\infty}{(c, c/(ab); q)_\\infty}$ for $\\abs{c/(ab)} \u003c 1$.\nIts terminating case leads to the [[q-Vandermonde identity|MBRSCQRGGH]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "EB4N9RQR51"
    ],
    "kind": "statement",
    "msc": [
      "33D15"
    ],
    "name": "Heine's transformation",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Basic_hypergeometric_series"
      }
    ],
    "tag": "A93A5VATSD",
    "uses": [
      "EB4N9RQR51",
      "Y1X8ZZQ7VM",
      "VVT27RB46B"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/A93A5VATSD/",
  "links": [
    {
      "field": "about",
      "target": "EB4N9RQR51"
    },
    {
      "field": "uses",
      "target": "EB4N9RQR51"
    },
    {
      "field": "uses",
      "target": "Y1X8ZZQ7VM"
    },
    {
      "field": "uses",
      "target": "VVT27RB46B"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/A93A5VATSD.md",
  "msc": [
    "33D15"
  ],
  "name": "Heine's transformation",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/A93/A93A5VATSD.md",
  "status": null,
  "tag": "A93A5VATSD",
  "uri": "m2://A93A5VATSD",
  "url": "https://tamnd.github.io/math2/wiki/heines-transformation/",
  "writing": {
    "tag": "A93A5VATSD",
    "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
      }
    ]
  }
}