Let $(X_{n})$ be a discrete time Markov chain with $E=\mathbb{N}$. Suppose that the transition matrix $P$ is given by:
$$P(n,n+1)=p(n), \quad P(n,n)=q(n) \quad \mbox{and} \quad P(n,n-1)=r(n), \quad r(0)=0$$
Let $a(z)=\mathbb{P}_{z}(\nu_{x}<\nu_{y})$, for all $x \leq z \leq y$, i.e. the probability that, starting at $z$ the state $x$ is reached before $y$. Define $$s(x)=\frac{r(1)\cdots r(x)}{p(1)\cdots p(x)}$$
Prove that $$a(z)=\frac{\sum_{n=z}^{y-1}s(n)}{\sum_{n=x}^{y-1}s(n)}$$
I have the recurrence relation $$a(n+1)=(a(1)-a(0))\sum_{n=1}^{n}s(n) +1$$, but I don't know how to use this to finish. If anybody could help me I would be very thankful.