@**article**{RISC3945,author = {Alin Bostan and Manuel Kauers},

title = {{The Complete Generating Function for Gessel Walks is Algebraic}},

language = {english},

abstract = { Gessel walks are lattice walks in the quarter plane $\set N^2$ which start at
the origin~$(0,0)\in\set N^2$ and consist only of steps chosen from the set
$\{\leftarrow,\penalty0\swarrow,\penalty0\nearrow,\penalty0\rightarrow\}$. We
prove that if $g(n;i,j)$ denotes the number of Gessel walks of length~$n$
which end at the point~$(i,j)\in\set N^2$, then the trivariate generating
series $\displaystyle\smash{ G(t;x,y)=\sum_{n,i,j\geq 0} g(n;i,j)x^i y^j t^n }
$ is an algebraic function.
},

journal = {Proceedings of the AMS},

volume = {138},

number = {9},

pages = {3063--3078},

isbn_issn = {ISSN 0002-9939},

year = {2010},

month = {September},

refereed = {yes},

length = {16}

}