Let $\Lambda$ be a lattice (i.e. $\Lambda \simeq \mathbb{Z}^n$) with a positive subcone $\Lambda^+$. Let $H: \Lambda^+ \rightarrow \mathbb{C}$ be a function such that for any $\mu \in \Lambda^+$, $b_\mu(\lambda) := \frac{H(\lambda + \mu)}{H(\lambda)}$ is a polynomial in $\lambda$. I think, but haven't figured out how to prove, the following:

$b_\mu$ factors as $b_\mu = \prod_i (\alpha_i + k_i)$, where $\alpha_i \in Hom(\Lambda, \mathbb{C})$.

In fact, I have a bigger conjecture:

$H(\lambda)$ is a product of factorials of linear factors (i.e. factors of the form $\alpha_i + k_i$ with $\alpha_i \in Hom(\Lambda, \mathbb{Z})$ multiplied by an exponential function multiplied by a product of linear factors.

A few examples: The most basic examples are products of factorials of linear factors; however, we can also consider the case $\Lambda = \mathbb{Z}, H(\lambda) = (3 \lambda)! (3 \lambda + 2)$.

I wasn't quite sure what to tag this with, so please feel free to tag it appropriately.