This question is a little bit of a fishing expedition. I have some functions defined in terms of sums over certain partitions (and motivated by geometry — specifically, Hilbert schemes of points on orbifold surfaces) that seem to interpolate between different modular forms and rational functions, and I’m hoping there’s a number-theoretic framework that they fit into.

These functions are of the form $ $ f(q,t)=\sum_{m,n\geq 0} a_{m,n}q^mt^n$ $ for nonnegative integers $ a_{m,n}$ that have interpretations as the dimensions of certain spaces, or as counts of partitions satisfying certain constraints.

For specific values of $ t$ , the $ f(q,t)$ seem to specialize to theta functions of lattice of different dimensions and to rational functions in $ q$ . In particular, it seems these functions have the following three specializations:

- $ f(q,0)$ is a theta function for a lattice of dimension $ d<2N$
- $ f(q, t^{-N-1/2})$ is the theta function for a lattice of dimension $ 2N$
- $ f(q,1)$ is a rational function, specifically: $ $ f(q,1)=\prod_{i=1}^N \frac{1}{(1-q^{c_i})}$ $ for some integers $ 1=c_1<c_2<\cdots <c_N<2N$

Point 2 I can prove in general, Point 3 is definitely true but I can’t prove it except in very special cases, and Point 1 is a bit speculative but something “close to it” is definitely true.

In the simplest case I can prove everything. Here we can write $ f(q,t)$ explicitly:

$ $ f(q,t)=\sum_{n=1}^\infty q^{n(n-1)/2} \sum_{k=0}^{n-1} q^kt^{k(n-k)}$ $

- At $ t=0$ , we have: $ $ f(q,0)=\sum_{n=1}^\infty q^{n(n-1)/2}=\prod_{m=1}^\infty\frac{(1-q^2)^{2m}}{(1-q^m)}$ $ is the generating function for triangular numbers (and 2-core partitions), the theta function for a 1-dimensional lattice
- At $ t=q^{-1-1/2}=q^{-3/2}$ , we have $ $ f(q,t^{-3/2})=\prod_{m=1}^\infty\frac{(1-q^{3m/2})^3}{(1-q^{m/2})}$ $ is the generating function for 3-core partitions evaluated $ q^{1/2}$ , instead of $ q$ , and the theta function of a 2-dimensional lattice. See A033687. Presumably you can see this specialization algebraically, but I only know it from the geometric context, and specifically I use a Riemann-Roch theorem to see this.
- At $ t=1$ , we have $ $ f(q,1)=\frac{1}{1-q}$ $

Any leads to related number theory, or explanation of why these aren’t so interesting, would be appreciated.