On the Holonomy or Algebraicity of Generating Functions Counting Lattice Walks in the Quarter-Plane

TL;DR

This paper investigates the nature of generating functions counting lattice walks in the quarter-plane, using algebraic and boundary value problem methods to determine their holonomy or algebraicity.

Contribution

It applies general theorems to directly analyze the generating functions, clarifying their algebraic or holonomic nature for certain lattice walk models.

Findings

Generating functions are holonomic for walks with finite group symmetries.

Some generating functions are algebraic, notably Gessel's walk.

The method provides a direct approach to determine the nature of these functions.

Abstract

In two recent works \cite{BMM,BK}, it has been shown that the counting generating functions (CGF) for the 23 walks with small steps confined in a quadrant and associated with a finite group of birational transformations are holonomic, and even algebraic in 4 cases -- in particular for the so-called Gessel's walk. It turns out that the type of functional equations satisfied by these CGF appeared in a probabilistic context almost 40 years ago. Then a method of resolution was proposed in \cite{FIM}, involving at once algebraic tools and a reduction to boundary value problems. Recently this method has been developed in a combinatorics framework in \cite{Ra}, where a thorough study of the explicit expressions for the CGF is proposed. The aim of this paper is to derive the nature of the bivariate CGF by a direct use of some general theorems given in \cite{FIM}.