The Question:
Is there any "well behaved" $H, \mathbb{C} \rightarrow \mathbb{C}$ such that the sum
$$ \sum_{a,b \in \mathbb{Z}} H(x+a+b\pi) $$
Converges for some $x$ in some connected subset of the complex plane.
Where this question came from:
Consider the functional equation looking for functions $f: \mathbb{C} \rightarrow \mathbb{C}$ such that
$$ f(x+1) + f(x+\pi) = f(x)$$
There is an trick that can be used here:
Which is we assume the form of the solutions is $ca^x$ for unknown complex constants $a,c \in \mathbb{C}$. Then we can solve:
$$ ca^{x+1} + ca^{x+\pi} = ca^x $$
Dividing through by $ca^x$ yields:
$$ a + a^{\pi} = 1 $$
And complex solutions to this give us solutions to the aforementioned equation (and can be combined through linear combinations to make more solutions).
But I want to approach this problem without any tricks or "intuition" and instead go purely from basic principles. As with any linear functional equation with constant coefficients we can suppose that the answer takes the form of an infinite sum as below:
$$ {\newcommand\iddots{\mathinner{ \kern1mu\raise1pt{.} \kern2mu\raise4pt{.} \kern2mu\raise7pt{\Rule{0pt}{7pt}{0pt}.} \kern1mu }}} f(x)= \left( \begin{array}{ccccccccc} \ddots & & \vdots & & \vdots && \vdots & \iddots \\ & & + & & + & & + & \\ ... & + & c_{-1,-1}H(x-1-\pi) & + & c_{-1,0}H(x-\pi) & + & c_{-1,1}H(x+1-\pi) & + & ...\\ & & + & & + & & + & \\ ... & + & c_{0,-1}H(x-1) & + & c_{0,0}H(x) & + & c_{0,1}H(x+1) & + & ... \\ & & + & & + & & + & \\ ... & + & c_{1,-1}H(x-1+\pi) & + & c_{1,0}H(x+\pi) & + & c_{1,1}H(x+1+\pi) & + & ... \\ & & + & & + & & + & \\ \iddots & & \vdots & & \vdots && \vdots & \ddots \end{array} \right) $$
Where $H: \mathbb{C} \rightarrow \mathbb{C}$ is some function you get to choose and the coefficients $c_{i,j}$ obey that $c_{i+1,j} + c_{i,j+1} - c_{i,j} = 0$
You can let $c_{n,n} = 1 \forall n \in \mathbb{Z}$ and let $c_{n,n-2} = 0 \forall n \in \mathbb{Z}$ and then using the relation you can fill in the rest of the table. Its possible to prove that the coefficients that result in the table from filling always are either $0,1,-1$ so really the question then is whether we find a suitable $H(x)$. In our case if we can find an $H(x)$ which converges when summed over the entire range of integer combination then it probably gives a hint as to what $H(x)$ should be for our more specific example.