Singularity analysis via the iterated kernel method

TL;DR

This paper proves that five specific lattice path models in the quarter plane have generating functions that are not D-finite, confirming a conjecture and providing exact and asymptotic formulas for their enumeration.

Contribution

It completes the proof that these five models' generating functions are not D-finite and offers explicit formulas for counting their walks.

Findings

Generated functions are not D-finite for the five models

Provided exact enumeration formulas for these models

Derived asymptotic estimates for the number of walks

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.