Hypergeometric expressions for generating functions of walks with small steps in the quarter plane

We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on Z2 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  N2, counted by their length and by the position of their ending point. Bousquet-Mélou and Mishna (2010) identified 19 models of walks that possess a D-finite generating function; linear differential equations have then been guessed in these cases by Bostan and Kauers (2009). We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss' hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their 19×4 combinatorially meaningful specializations only four are algebraic functions.