In branches of algebra impinging on the enumeration of partitions, one often encounters formulas like

$$\prod_i \left( \frac{1}{1-q^i} \right)^{n_i}$$

for some integers $n_i$. E.g., with $n_i = 1$, this counts partitions, and with $n_i = i$, plane partitions.

Such formulas are usually understood as formal products, but if you tried to take them seriously as analytic functions, you would be thinking about functions with poles at all roots of unity.

Is there a branch of analysis which studies such functions? Does that theory have consequences in combinatorics?

(I don’t know even enough analysis to know what tag to put.)