Consider $P(a,b)$ to be a multivariate probability mass function with discrete support. In particular $a,b \in 0,1,2,3 ..$
Let me define a probability generating function as follows:
\begin{equation} Q_a(z) = \sum_b z^b P(a,b) \end{equation}
and suppose that such function satisfies this functional equation: \begin{equation} Q_a(z)[(z-1)\gamma\cdot a]=0 \end{equation}
where $\gamma \in R^+$.
Is there any method to try to approach this problem and hence compute $Q_a(z)$? Or is there any known literature on such functional equations?