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  "body": "\n## Definition\n\nLet $\\sum_{n \\ge 0} c_n$ be a series of nonzero complex numbers.\nView the term ratio $c_{n+1} / c_n$ as a function of $n$ extended to complex values.\nThe series is an *elliptic hypergeometric series* if this ratio is an [[elliptic function|FF9ZPCZ59K]] of $n$.\n\nIn the usual explicit form, fix complex numbers $p, q$ with $\\abs{p} \u003c 1$ and $q \\ne 0$.\nThe *modified theta function* is $\\theta(x; p) = (x; p)_\\infty \\, (p/x; p)_\\infty$ for $x \\ne 0$, where $(\\cdot\\,; p)_\\infty$ is the infinite [[q-Pochhammer symbol|Y1X8ZZQ7VM]].\nThe *elliptic shifted factorial* is $(a; q, p)_n = \\prod_{k=0}^{n-1} \\theta(a q^k; p)$.\nFor nonzero parameters $a_1, \\ldots, a_{r+1}$, $b_1, \\ldots, b_r$ and $z$, the series\n$${}_{r+1}E_r(a_1, \\ldots, a_{r+1}; b_1, \\ldots, b_r; q, p; z) = \\sum_{n \\ge 0} \\frac{(a_1; q, p)_n \\cdots (a_{r+1}; q, p)_n}{(q; q, p)_n \\, (b_1; q, p)_n \\cdots (b_r; q, p)_n} z^n$$\nis an elliptic hypergeometric series when the balancing condition $a_1 a_2 \\cdots a_{r+1} = q \\, b_1 b_2 \\cdots b_r$ holds.\n\n## Remarks\n\nWhy the balancing condition: write $q = e^{2\\pi i \\sigma}$ and $p = e^{2\\pi i \\tau}$, and view the term ratio as a function of a complex variable $n$.\nThe shift $n \\mapsto n + 1/\\sigma$ leaves $q^n$ unchanged.\nThe shift $n \\mapsto n + \\tau/\\sigma$ replaces $q^n$ by $p q^n$, and the rule $\\theta(px; p) = -x^{-1}\\theta(x; p)$ shows that it multiplies the ratio by $q b_1 \\cdots b_r / (a_1 \\cdots a_{r+1})$.\nSo the ratio is elliptic with periods $1/\\sigma$ and $\\tau/\\sigma$ exactly when this factor is $1$.\n\nExample: for $p = 0$ one has $\\theta(x; 0) = 1 - x$ and $(a; q, 0)_n = (a; q)_n$, so the series becomes the [[basic hypergeometric series|EB4N9RQR51]] ${}_{r+1}\\phi_r$ with the same parameters.\n\nNon-example: $\\sum_n n! \\, z^n$ has term ratio $(n + 1) z$, a nonconstant polynomial in the complex variable $n$.\nA nonconstant elliptic function must have poles, so the series is not elliptic hypergeometric.\n",
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