Type: Article
Publication Date: 2010-09-01
Citations: 120
DOI: https://doi.org/10.1090/s0002-9939-2010-10398-2
Gessel walks are lattice walks in the quarter-plane $\mathbb N^2$ which start at the origin $(0,0)\in \mathbb N^2$ and consist only of steps chosen from the set $\{\leftarrow , \swarrow , \nearrow , \rightarrow \}$. We prove that if $g(n;i,j)$ denotes the number of Gessel walks of length $n$ which end at the point $(i,j)\in \mathbb N^2$, then the trivariate generating series $\displaystyle {G(t;x,y)=\sum _{n,i,j\geq 0} g(n;i,j)x^i y^j t^n}$ is an algebraic function.