Consider a uniform $\{m,n\}$ tiling of the hyperbolic plane, for convenience with one vertex at the origin (and also for convenience, normalize the edges to have unit hyperbolic length). Then there are (at least) two different notions of a 'theta series' for this tiling that make sense: one is the sum $\sum_{v}q^{|v|^2}$ for all vertices $v$ of the tiling, where $|v|^2$ is the square of the (hyperbolic) distance from $v$ to the origin; the other would be a sum along the lines of $\sum_{v}q^{|v|_G}$ where $|v|_G$ is the 'graph distance' between $v$ and the origin; that is, the length of the shortest path of edges that connects $v$ to the origin.
The former may be an awkward object of study, because while it's the closer analog to a Euclidean theta-function, unless I'm mistaken most of the exponents are going to be transcendental, of the form $\log \alpha$ for some algebraic (rational?) number $\alpha$, and these are unlikely to coalesce nicely. OTOH, the graph version, while it's less geometrically motivated, is much more manageable; certainly in the $\{5,4\}$ case I'm pretty sure I can prove that it's a rational function, and I suspect that that's true in the general case as well. (The coefficients look to satisfy a relatively straightforward recurrence relation that would imply rationality).
I'm wondering whether either of these objects has been studied before, and what's known about them; in particular, whether the 'geometric distance' theta series is in fact as awkward as it looks or whether there's some clever transformation to make it manageable, and whether a general-purpose explicit expression is known in terms of $m$ and $n$ for the 'graph distance' series.