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For $|x|<1$, and states $i,j\in S$ define: $$ P_{ji} (x)=\sum_{n=0}^{\infty} p_n(j|i) x^n $$ $$ F_{ji} (x)=\sum_{n=1}^{\infty} f_n(j|i) x^n $$ Where $f_n (j|i)=P(\xi_n=j,\xi_k\neq j,k=1..n-1|\xi_0=i)$ (i.e. that the state j first occurs at step n) and $p_n(j|i)=P(\xi_n=j|\xi_0=i)$. Prove the following relations: $$ P_{ji} (x)=F_{ji} (x)P_{jj} (x), j\neq i $$ $$ P_{ii} (x)=1+F_{ii} (x)P_{ii} (x) $$ This is problem 5.17 of the book Basic Stochastic Processes by Brzezniak & Zastawniak. The solutions for this question contains the proof that $$ p_n(j|i)= \sum_{k=1}^{n} f_k(j|i) p_{m-k}(j|j) $$ and claims this is sufficient to prove the above relations. I can derive from that the following: $$ P_{ji} (x)=\sum_{n=0}^{\infty} p_n(j|i) x^n =\sum_{n=0}^{\infty} \left(\sum_{k=1}^{n} f_k(j|i) p_{n-k}(j|j)\right) x^n $$ $$ =\sum_{n=0}^{\infty} \sum_{k=1}^{n} f_k(j|i) p_{n-k}(j|j) x^n $$ but I'm at a loss for where to go from here. How can I get from this to the sum for $F_{ji} (x)P_{jj} (x)$?

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