I have an infinite sum written as

$$

\sum_{mn} e^{-i2\pi(m c_1 +n c_2)} e^{-(m^2+n^2-mn)}

$$

where $m,n$ are integers, $0<c_1, c_2<1$.

I want to express the above expression into a product of two theta functions. If I don't have the cross term $e^{-mn}$, it seems straightforward.

## Best Answer

I can get you a sum of two products of theta functions (with a prefactor). To clear the cross term, make the substitution $m \mapsto u+v$, $n \mapsto u-v$. Sum that and then, to your original sum, make the substitution $m \mapsto u+v+1$, $n \mapsto u-v$ and sum that. Then, adding the two together, ...

where $\vartheta_3(u ; q) = 1 + 2 \sum_{n=1}^\infty q^{n^2} \cos(2 n u)$ (which I only write down because nobody can settle on notation for theta functions).