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  "body": "\n## Definition\n\nLet $p, q$ be complex numbers with $\\abs{p} \u003c 1$ and $\\abs{q} \u003c 1$.\nFor $z \\in \\C \\setminus \\set{0}$ with $p^j q^k z \\ne 1$ for all integers $j, k \\ge 0$, the *elliptic gamma function* is\n$$\\Gamma(z; p, q) = \\prod_{j, k \\ge 0} \\frac{1 - p^{j+1} q^{k+1} / z}{1 - p^j q^k z}.$$\n\n## Remarks\n\nWith the theta function $\\theta(z; p) = (z; p)_\\infty\\, (p/z; p)_\\infty$, built from the [[q-Pochhammer symbol|Y1X8ZZQ7VM]] $(a; p)_\\infty = \\prod_{j \\ge 0}(1 - a p^j)$, the function satisfies\n$$\\Gamma(qz; p, q) = \\theta(z; p)\\, \\Gamma(z; p, q).$$\nCheck: in the quotient $\\Gamma(qz)/\\Gamma(z)$ all factors cancel except the factors with $k = 0$, which give $\\prod_{j \\ge 0}(1 - p^{j+1}/z)(1 - p^j z) = \\theta(z; p)$.\n\nExample: replacing $z$ by $pq/z$ exchanges the numerator and the denominator of the product, so $\\Gamma(z; p, q)\\, \\Gamma(pq/z; p, q) = 1$.\n\nNon-example of a valid argument: $z = 1$ makes the factor $1 - p^0 q^0 z$ vanish, so $\\Gamma(z; p, q)$ has a pole at $z = 1$.\n\nThe function is symmetric in $p$ and $q$.\n",
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