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 $a\in\mathbb{N}$.

A vast generalization of Weierstraß's theorem is to Riemann surfaces.
Florack, inspired by methods due to Behnke-Stein,  proved the following in 1948:  
Let $X$ be a non-compact Riemann surface. Let $D$ be a closed discrete set in $X$ and to each $d\in D$ attach a complex number $a_d$.
 Then there exists a holomorphic function $f\in \mathcal O(X)$ defined on all of $X$ such that  $f(a_d)=c_d.$
One may think of $f$ as holomorphically  interpolating  some discrete data.
This result immediately implies that $X$ is a Stein manifold, a concept of fundamental importance in the theory of holomorphic manifolds: Stein manifolds are the analogues  of affine varieties in algebraic geometry. 
A complete proof is in Theorem 26.7 of  Forster's awesome Lectures on Riemann Surfaces.
The special case where $X$ is an open subset of $\mathbb C$  is analyzed in John's answer.
