Elliptic hypergeometric series

Definition

Let $\sum_{n \ge 0} c_n$ be a series of nonzero complex numbers. View the term ratio $c_{n+1} / c_n$ as a function of $n$ extended to complex values. The series is an elliptic hypergeometric series if this ratio is an elliptic function of $n$.

In the usual explicit form, fix complex numbers $p, q$ with $\abs{p} < 1$ and $q \ne 0$. The 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. The elliptic shifted factorial is $(a; q, p)_n = \prod_{k=0}^{n-1} \theta(a q^k; p)$. For nonzero parameters $a_1, \ldots, a_{r+1}$, $b_1, \ldots, b_r$ and $z$, the series $${}_{r+1}E_r(a_1, \ldots, a_{r+1}; b_1, \ldots, b_r; q, p; z) = \sum_{n \ge 0} \frac{(a_1; q, p)_n \cdots (a_{r+1}; q, p)_n}{(q; q, p)_n \, (b_1; q, p)_n \cdots (b_r; q, p)_n} z^n$$ is 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.