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4 I want to determine the generating function of the numbers of visits up to a specific time. Formally: let ${X_{n}}_{n geq 0}$ be a markov chain on the state space $S = {1,ldots,r}$ with transition matrix $P = (p_{i,j})_{1 leq i,j leq r}$ and initial probability vector $alpha = (alpha_{i} : i in S)$ with $alpha_{i} = mathbb{P}(X_{0} = i)$ . Let $q(n_{1},ldots,n_{r})$ be the probability that up to time $n-1$ the state $1$ is visited $n_{1}$ -times, $ldots$ ,state $r$ is visited $n_{r}$ -times and $F_{n}(x_{1},ldots,x_{r}) = sum_{n_{1}+ cdots +n_{r} = n}q(n_{1},ldots,n_{r})x_{1}^{n_{1}}cdots x_{r}^{n_{r}}$ be the corresponding generating function of $q(n_{1},ldots,n_{r})$ . Now I want to show that $$ F_{n}(x_{1},ldots,x_{r}) = (alpha_{1}x_{1},ldots,alpha_{r}x_{r}) left( P D_{r}right)^{n-1} e, $$ where $e = (