Immigration Plus Branching in One Step

Let $Z_t$ be a branching process with immigration. Each individual in generation $t$ produces offspring with PGF $\phi(s)$, independently, and the number of immigrants arriving at the next generation has PGF $\psi(s)$, independently of everything else. Express the PGF of $Z_{t+1}$ in terms of the PGF of $Z_t$.