I'm trying to find the probability generating function of a branching process with immigration that can be described as the following:
$$Z_{n+1} = I_n + \sum_{i = 0}^{Z_n}Y_{n,i}$$
Clearly, we can write the pgf as: $$G_{Z_{n+1}}(s) = G_I(s) \times G_{Z_n}(G_Y(s))$$
We can continue this further inductively: $$G_{Z_{n+1}}(s) = G_I(s) \times G_{I}(G_Y(s)) \times G_{Z_{n-1}}\circ G_Y \circ G_Y(s)$$ $$G_{Z_{n+1}}(s)= G_I(s) \times G_{I}\circ G_Y(s) \times G_I \circ G_Y \circ G_Y(s) \times G_{Z_{n-2}}\circ G_Y \circ G_Y \circ G_Y(s)$$
But I'm having trouble simplifying this and writing it as a simple equation in terms of $G_I, G_Y$ and $n$. Have I missed something or is my approach completely wrong?