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  "body": "\n## Statement\n\nLet $q, z \\in \\C$ with $\\abs{q} \u003c 1$ and $z \\ne 0$.\nThen\n$$\\prod_{m=1}^{\\infty} \\left(1 - q^{2m}\\right) \\left(1 + z q^{2m-1}\\right) \\left(1 + z^{-1} q^{2m-1}\\right) = \\sum_{n=-\\infty}^{\\infty} q^{n^2} z^n.$$\nEquivalently, with the [[q-Pochhammer symbol|Y1X8ZZQ7VM]], $(q; q)_\\infty\\, (z; q)_\\infty\\, (q/z; q)_\\infty = \\sum_{n \\in \\Z} (-1)^n q^{\\binom{n}{2}} z^n$.\n\n## Remarks\n\nProof idea: call the product $F(z)$; it is holomorphic on $\\C \\setminus \\set{0}$, so it has a Laurent expansion $\\sum_n c_n z^n$.\nThe product satisfies $F(q^2 z) = (q z)^{-1} F(z)$, which forces $c_n = q^{n^2} c_0$.\nThe constant $c_0 = 1$ can be found by specializing $z$, for instance with Gauss's or Euler's identities, or by a limit argument.\nExample: substituting $q \\mapsto q^{3/2}$ and $z \\mapsto -q^{1/2}$ in the first form gives [[Euler's pentagonal number theorem|R587SR9GNV]] $(q; q)_\\infty = \\sum_{n \\in \\Z} (-1)^n q^{n(3n-1)/2}$.\nWith $z = 1$ the first form gives $\\sum_{n \\in \\Z} q^{n^2} = \\prod_{m \\ge 1} (1 - q^{2m})(1 + q^{2m-1})^2$, a product for a [[theta function|2F5CYQJVAF]].\nThe condition $\\abs{q} \u003c 1$ is needed for the [[infinite product|RZ0Z2BTMT4]] and the series to converge.\n\n## Explanations\n\n### L1 human\n\nThe triple product identity says that a certain infinite product equals a sum of powers $q^{n^2}$, one for each integer $n$.\nThe sum side is a theta function, so the identity turns theta functions into products.\nProducts are good for counting partitions, sums are good for computing, and the identity lets one pass between them.\n\n### L2 human\n\nIt is the basic identity behind the product formulas for the Jacobi theta functions and the [[Dedekind eta function|MX3CVYMNDV]].\nIt is a limiting case of [[Ramanujan's 1psi1 summation|RK6DCJVJCR]].\nTogether with its relative, the [[quintuple product identity|PZR6RC79ED]], it gives many partition identities, and it is a step in proofs of the [[Rogers–Ramanujan identities|M24H4GJPPW]].\nIn Lie theory it is the Macdonald identity for the affine root system of type $A_1^{(1)}$.\n",
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