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 $\forall\mu \in \Lambda^+$, $b_\mu(\lambda) := \frac{H(\lambda + \mu)}{H(\lambda)}$ is a polynomial in $\lambda$. To put it another way, let $b: \Lambda \rightarrow Rat(\Lambda)$ be a cocycle (in the group cohomology complex with $Rat(\Lambda)$ being rational functions on $\mathbb{C}\Lambda$ and the translation action) such that any element of the positive subcone is a polynomial. I think, but haven't figured out how to prove, the following:
$$b_\mu ~~\text{factors as}~~ b_\mu = \prod_i (\alpha_i + k_i), ~~\text{where}~~ \alpha_i \in \text{Hom}(\Lambda, \mathbb{C})$$
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 $\alpha_i \in \text{Hom}(\Lambda, \mathbb{Z})$ 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.
