Proof of a Conjecture on the Slit Plane Problem

Let $a_{i,j}(n)$ denote the number of walks in $n$ steps from $(0,0)$ to $(i,j)$, with steps $(\pm 1,0)$ and $(0,\pm 1)$, never touching a point $(-k,0)$ with $k\ge 0$ after the starting point. \bous and Schaeffer conjectured a closed form for the number $a_{-i,i}(2n)$ when $i\ge 1$. In this paper, we prove their conjecture, and give a formula for $a_{-i,i}(2n)$ for $i\le -1$.