Singularity analysis via the iterated kernel method

Abstract

In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach of Fayolle, Rachel, and Kurkova. Here we consider these five models, called the singular models, and prove that the generating functions marking the number of walks of a given length are not D-finite -- thus finishing the proof of a conjecture of Bousquet-M\'elou and Mishna. Furthermore, we provide exact and asymptotic enumerative formulas for the number of such walks.