Let $X,Y$ be two random variables. We denote by $[X^k]$ and $[Y^k]$ the $k$'th order cumulants of $X$ and $Y$, respectively. I'm interested in computing the $k$'th order cumulant of $Z = X+Y$.
If $X,Y$ were independent, then a well-known property implies that $[Z^k] = [X^k] + [Y^k]$.
Now suppose $X,Y$ are not independent. Can we write an expression for the cumulants of $Z$, from the cumulants of $X$, $Y$, plus additional terms related to the dependence between $X$ and $Y$ (e.g., the cumulants of products $XY$?
Update: wolfie's post almost answers this question. This expansion feels like something that should be known in the literature. So I am adding the
reference tag here, in case anyone can suggest relevant papers.