Two non-holonomic lattice walks in the quarter plane

We present two classes of random walks restricted to the quarter plane with non-holonomic generating functions. The non-holonomicity is established using the iterated kernel method, a variant of the kernel method. This adds evidence to a recent conjecture on combinatorial properties of walks with holonomic generating functions [M. Mishna, Classifying lattice walks in the quarter plane, J. Combin. Theory Ser. A 116 (2009) 460–477]. The method also yields an asymptotic expression for the number of walks of length n.