Elliptic gamma function

Remarks

With the theta function $\theta(z; p) = (z; p)_\infty\, (p/z; p)_\infty$, built from the q-Pochhammer symbol $(a; p)_\infty = \prod_{j \ge 0}(1 - a p^j)$, the function satisfies $$\Gamma(qz; p, q) = \theta(z; p)\, \Gamma(z; p, q).$$ Check: in the quotient $\Gamma(qz)/\Gamma(z)$ all factors cancel except the factors with $k = 0$, which give $\prod_{j \ge 0}(1 - p^{j+1}/z)(1 - p^j z) = \theta(z; p)$.

Example: replacing $z$ by $pq/z$ exchanges the numerator and the denominator of the product, so $\Gamma(z; p, q)\, \Gamma(pq/z; p, q) = 1$.

Non-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 $z = 1$.

The function is symmetric in $p$ and $q$.