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Suppose I have two independent integer random variables $X_1$, $X_2$ (with constraint that $X_1+X_2\le N,0\le X_1\le N,0\le X_2\le N$), with probability generating functions $g_1(z)$, $g_2(z)$. Now I have a joint-distribution $P(X_1-X_2,2X_1+X_2-N)$, whose probability generating function is $G(z_1,z_2)$. $N$ is a constant.
What is the relation between $G(z_1,z_2)$ and $g_1(z)$, $g_2(z)$?
The following question is related but not the same: multivariate probability generating function
Note that $$ G(z_1,z_2)=E(z_1^{X_1-X_2}z_2^{2X_1+X_2-N})=z_2^{-N}E((z_1z_2^2)^{X_1}(z_1^{-1}z_2)^{X_2}). $$ Since $X_1$ and $X_2$ are independent, $$ G(z_1,z_2)=z_2^{-N}g_1(z_1z_2^2)g_2(z_1^{-1}z_2). $$ This does not use the constraints.
