Counting Walks in the Quarter Plane

We study planar walks that start from a given point (i\_0, j\_0), take their steps in a finite set S, and are confined in the first quadrant of the plane. Their enumeration can be attacked in a systematic way: the generating function Q(x, y, t) that counts them by their length (variable t) and the coordinates of their endpoint (variables x, y) satisfies a linear functional equation encoding the step-by-step description of walks. For instance, for the square lattice walks starting from the origin, this equation reads (xy-t(x + y + x^2 y + x y^2)) Q(x, y, t) = xy-xtQ(x, 0, t)-ytQ(O, y, t). The central question addressed in this paper is the nature of the series Q(x, y , t). When is it algebraic? When is it D-finite (or holonomic)? Can these properties be derived from the functional equation itself? Our first result is a new proof of an old theorem due to Kreweras, according to which one of these walk models has, for mysterious reasons, an algebraic generating function. Then, we provide a new proof of a holonomy criterion recently proved by M. Petkovsek and the author. In both cases, we work directly from the functional equation.