On the functions counting walks with small steps in the quarter plane

Models of spatially homogeneous walks in the quarter plane $\mathbf{ Z}_{+}^{2}$ with steps taken from a subset $\mathcal{S}$ of the set of jumps to the eight nearest neighbors are considered. The generating function (x,y,z)↦Q(x,y;z) of the numbers q(i,j;n) of such walks starting at the origin and ending at $(i,j) \in\mathbf{ Z}_{+}^{2}$ after n steps is studied. For all non-singular models of walks, the functions x↦Q(x,0;z) and y↦Q(0,y;z) are continued as multi-valued functions on C having infinitely many meromorphic branches, of which the set of poles is identified. The nature of these functions is derived from this result: namely, for all the 51 walks which admit a certain infinite group of birational transformations of C2, the interval $]0,1/|\mathcal{S}|[$ of variation of z splits into two dense subsets such that the functions x↦Q(x,0;z) and y↦Q(0,y;z) are shown to be holonomic for any z from the one of them and non-holonomic for any z from the other. This entails the non-holonomy of (x,y,z)↦Q(x,y;z), and therefore proves a conjecture of Bousquet-Mélou and Mishna in Contemp. Math. 520:1–40 (2010).