Probability generating function for sum of non-independent random variables

Question

Given two random variables $X,Y$, generated by pgfs $g_X,g_Y$, their sum $Z=X+Y$ is generated by $g_Z(s)=g_X(s)g_Y(s)$ if $X$ and $Y$ are independent. Is it possible to find the pgf of $Z$ when we don't know that $X$ and $Y$ are independent?

Other questions (this one, another) deal with the case where $X$ and $Y$ are independent, and I haven't been able to find a question that matches my case.

Answer 1

If they are not independent, you need to know the joint distribution of $X$ and $Y$. It is not determined by the pgf's of $X$ and $Y$.

Answer 2

From the joint distribution of $(X,Y)$, which indeed you need to know, you obtain the probability generating function $g_{X,Y}(s_1,s_2)$. Then the probability generating function of $X+Y$ is simply $g_{X,Y}(s,s)$.