multivariate probability generating function

Question

Suppose I have three random variables $X_1$, $X_2$ and $X_3$, with probability generating functions $g_1(z)$, $g_2(z)$ and $g_3(z)$. Now I have a joint-distribution $P(X_1-X_2,X_1-X_3)$, whose probability generating function is $G(z_1,z_2)$.

What is the relation between $G(z_1,z_2)$ and $g_1(z)$, $g_2(z)$, $g_3(z)$?

Answer 1

If $(X_1,X_2,X_3)$ is independent, $$ G(z_1,z_2)=E(z_1^{X_1-X_2}z_2^{X_1-X_3})=E((z_1z_2)^{X_1}z_1^{-X_2}z_2^{-X_3}), $$ hence $$ G(z_1,z_2)=g_1(z_1z_2)g_2(z_1^{-1})g_3(z_2^{-1}). $$