Section: New Results

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

In [2], Alin Bostan and Frédéric Chyzak, together with Mark van Hoeij (Florida State University), Manuel Kauers (Johannes Kepler University), and Lucien Pech, have studied 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 consists of non-zero vectors with coordinates 0, 1 or $-1$. They concerned themselves 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. In earlier works, Bousquet-Mélou and Mishna had identified 19 models of walks that possess a D-finite generating function, and linear differential equations had then been guessed in these cases by Bostan and Kauers. Here, we have given the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we have proved that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions, with specific parameters that relate them intimately to elliptic integrals. As a second corollary, we have shown that all the 19 generating functions are transcendental, and that among their $19\times 4$ combinatorially meaningful specializations only four are algebraic functions.