My question concerns Exercise 1.6.47. from Pierre Brémaud's (2020, p. 49-50) Probability Theory and Stochastic Processes.
We are given a recursively defined sequence $\{{X_{n}}\}_{n≥0}$ where $ X_{n+1} = max(X_{n}-1,0) + Z_{n+1}$ $(n ≥ 0)$, and $X_0$ is a random variable taking its values in $\mathbb{N}$, and $\{{Z_{n}}\}_{n≥1}$ is a sequence of independent random variables taking their values in $\mathbb{N}$, and independent of $X_0$.
We are asked to express the generating function $ψ_{n+1}$ of $X_{n+1}$ in terms of the generating function $\phi$ of $Z_{1}$.
My approach is to take the expectation of $z^{X_{n+1}}$ and exploit independence with the $Z_{n+1}$ to give:
\begin{equation}\label{eq:psi recursive} \psi_{n+1}(z) = E[z^{max(X_{n}-1,0)}]\phi(z)\\ = \displaystyle \sum_{x = 0}^{\infty}z^{x}P(max(X_{n}-1,0)=x)\phi(z)\\ = \displaystyle [P(X_{n}=0)+P(X_{n}=1)]\phi(z)+\sum_{x = 1}^{\infty}z^{x}P(X_{n}=x+1)\phi(z) \end{equation}
The above leads to a recursive expression for $ψ_{n+1}$ in terms of $ψ_{n}$, $ψ_{n}(0)$ and $\phi$. However, I am stuck at this point because it seems the question wants us to find an explicit expression for $\psi_{n+1}$, but I don't see how to proceed due to the appearance of the $\psi_{n}(0) (= P(X_{n}=0))$ terms in the recursive definition of $\psi_{n+1}$.
Am I missing something or is this all there is to the question?
Any hints would be appreciated.
Thanks in advance.