recurrent states in markov chain with poisson-like transition matrix I am considering a Markov chain $X$ with state space $\mathbb{N}$ that has transition probabilities $p_{ij}=\begin{cases}1\mbox{ for }i=j=0\\e^{-i}\frac{i^{j}}{j!} \mbox{ otherwise }\end{cases}.$ I want to identify the recurrent states of the chain (other than the state $0$). I know that a state $i$ is recurrent iff $\sum_{n=1}^{\infty}\mathbb{P}(X_{n}=i|X_{0}=i)=\infty.$ Expanding the LHS using the Markov property gives the expression $$\sum_{n=1}^{\infty}\sum_{k_1}\ldots\sum_{k_n}p_{k_{1}i}p_{k_{2}k_{1}}\ldots p_{ik_{n}}=\sum_{n=1}^{\infty}\sum_{k_1}\ldots\sum_{k_n}e^{-k_1}\frac{k_{1}^{i}}{i!}e^{-k_2}\frac{k_{2}^{k_1}}{k_{1}!}\ldots e^{-i}\frac{i^{k_n}}{k_{n}!}.$$ Is there a way to calculate this series explicitly (or, in the case of a recurrent state, to show that it diverges)?
 A: This is a branching process in disguise since, conditionally on $X_n$, $X_{n+1}$ is distributed as the sum of $X_n$ independent copies of a random variable $L$ with Poisson distribution of parameter $1$. Since $E[L]=1$, the process is critical hence $T_0=\inf\{n\geqslant1\mid X_n=0\}$ is almost surely finite with respect to $P[\ \mid X_0=i]$ for every $i\geqslant0$. Thus, the process is absorbed at $0$ after a finite time, almost surely.
Furthermore, for every $i\geqslant1$, the number of visits $N_i=\sum\limits_{n=1}^{+\infty}\mathbf 1_{X_n=i}$ of the state $i$ is geometrically distributed and integrable with respect to $P[\ \mid X_0=i]$. In particular, for every $i\geqslant1$, the series $s(i)=\sum\limits_{n}P[X_n=i\mid X_0=i]$ converges.
Note that $s(i)=E_i[N_i]$ hence $s(i)=1/P[T_i=\infty\mid X_0=i]$, that is, $s(i)=1/(1-u_i)$ where $u_k=P[T_i\lt\infty\mid X_0=k]$. Thus, $u_0=0$ and, for every $k\ne i$, $u_k=\mathrm e^{-k}\frac{k^i}{i!}+\mathrm e^{-k}\sum\limits_{n\ne i}\frac{k^n}{n!}u_n$.
Whether one can deduce an explicit formula for $s(i)$ from this remains to be seen.
