What are the modular transformation properties of q-Pochhammer symbols? Do q-Pochhammer symbols, defined as
$(a;q) = \prod_{k=0}^{\infty} (1- a q^k)$ 
have known modular transformation properties?  That is, if we write $q = q[z] = e^{2\pi i z}$, is there any reasonably simple relationship between e.g. $(a;q[z])$ and $(a; q[-1/z])$ with $z$ in the upper half plane?
 A: When $a=1$, the transformation formula for the Dedekind eta function implies that $(1;q[-\frac{1}{z}])=e^{(z+\frac{1}{z})(i\pi)/12}\sqrt{-iz}(1;q[z])$ for $z$ in the upper half-plane. For $a=-1$, the identity $(-1;q[z])=\frac{(1;q[2z])}{(1;q[z])}$ together with the previous formula gives a relation, which means that $(-1;q[z])$ is essentially a modular function. However, for $a\neq \pm 1$, there at least cannot be any simple ''multplicative'' relation between $(a;q[z])$ and $(a;q[-\frac{1}{z}])$ since they have very different zeros: if $a=re^{i\varphi}$, the first one has zeros at $z=\frac{1}{2\pi i k}\log \frac{1}{r}-\frac{\varphi}{2\pi  k}+\frac{m}{ik}$, while the second one has zeros at $z=-(\frac{1}{2\pi i k}\log \frac{1}{r}-\frac{\varphi}{2\pi k}+\frac{m}{ik})^{-1}$, and it is pretty clear that these are different sets of zeros. 
A: If you wish to know an expansion of the q-Pochhammer symbol around $z\sim 0$ then there is
such an expansion.
$q=e^{-z}$
$(q\, a; q) =\exp\left(-\sum_{\ell=0}^\infty \frac{B_{\ell}}{\ell!} z^{\ell-1} \, Li_{2-\ell}(a)\right)$
Here $B_\ell$ are Bernoulli numbers and $Li_\ell(z)$ is polylogarithm. 
