Weierstrass Factorization Theorem Are there any generalisations of the Weierstrass Factorization Theorem, and if so where can I find information on them? I'm trying to investigate infinite products of the form
$$\prod_{k=1}^\infty f(z)^{k^a}e^{g(z)},$$
where $g\in\mathbb{Z}[z]$ and $a\in\mathbb{N}$.

The Weierstrass factorization theorem provides a way of constructing an entire function with any prescribed set of zeros, provided the set of zeros does not have a limit point in $\mathbb{C}$. I know that this generalizes to being able to construct a function holomorphic on a region $G$ with any prescribed set of zeros in $G$, provided that the set of zeros does not have a limit point in $G$. 
These are theorems VII.5.14 and VII.5.15 in Conway's Functions of One Complex Variable. They lead to the (important) corollary that every meromorphic function on an open set $\Omega$ is a ratio of functions holomorphic on $\Omega$.
