# Repeated reflection

This is a problem from Feller's book introduction to probability theory and its application, Vol 1, Chap 3 problem 3.

Let $a$ and $b$ be positive, and $-b <c<a$. Prove the number of paths to the point $(n,c)$ which meet neither the line $x=-b$ nor $x=a$ is given by the series $$\tag1 \sum_{k={-\infty}}^{\infty} (N_{n,4k(a+b)+c}-N_{n,4k(a+b)+2a-c})$$ where $N_{n,k}$ is the number of paths going from origin to point $(n,k)$ and only finitely many terms in $(1)$ are non-zero.

The hint suggests to use the reflection repeatedly, but why $4k(a+b)+c$ not $2k(a+b)+c$ ? Is this a typo in the problem?

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I haven't looked at your problem but Feller's texts both volumes as brilliant as they are are load with little errors. So my apriori guess based on that knowledge is that you could be right@ –  Michael Chernick Sep 8 '12 at 0:35