Hypergeometric Expressions for Generating Functions of Walks with Small Steps in the Quarter Plane

Understand

We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on $\mathbb{Z}^2$ defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or $-1$.

• We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane $\mathbb{N}^2$, counted by their length and by the position of their ending point.
• Bousquet-M\'elou and Mishna [Contemp.
• Math., pp.