I was wondering there there are any pairs of functions $(f,g)$ such that $$\sum_{n=1}^{\infty} (f \circ g)(n) = \sum_{n=1}^{\infty} (g \circ f)(n) $$ on condition that they're not commutative with respect to composition, i.e. $f \circ g \neq g \circ f $ for all $n$.
I must admit that I don't have a very clear motivation for this question, other than curiosity and that it seemed like a challenging though interesting problem on its own. The sum may also go over $ -\infty$ to $\infty$.
If any research has been done on this problem already, I'd like to learn more about it.