Compute a delta function summation over two integers $(-1)^{nm}\delta(x-an-bm)$

Question

I would like to evaluate the following summation over two integer variables $n,m$

$$\sum_{n,m}(-1)^{nm}\delta(x-an-bm)$$

where $a,b\in\mathbb R$ can be any real number

Is there some way this could be simplified to only sum over one of the integer variables?

Answer 1

For any order of summation, your series will diverge in the sense of distributions: choosing an enumeration $n_j,m_j$ of the indices and a test function such that $\phi(0)\ne 0$

$$\lim_{J\to \infty} \langle \sum_{j=1}^J(-1)^{n_jm_j}\delta(x-a n_j +bm_j),\phi \rangle=\sum_{j=1}^\infty (-1)^{n_jm_j}\phi(a n_j +bm_j)$$ There are infinitely many $a n_j +bm_j$ arbitrary close to $0$, for those $j$ then $\phi(a n_j +bm_j)$ is arbitrary close to $\phi(0)$ so the series $\sum_{j=1}^\infty (-1)^{n_jm_j}\phi(a n_j +bm_j)$ has infinitely many terms of absolute value $>\frac12|\phi(0)|$.

I doubt you have any good reason to look at this kind of series.