Walks in the quarter plane, genus zero case

TL;DR

This paper investigates the generating series of weighted quarter-plane walks with genus zero kernel curves, proving they do not satisfy any nontrivial algebraic differential equations with rational coefficients using Galois theory of difference equations.

Contribution

It introduces a Galois-theoretic approach to analyze the differential properties of generating series for genus zero quarter-plane walks, establishing non-algebraic differential nature.

Findings

Generating series are not solutions to any nontrivial algebraic differential equations.

Galois theory of difference equations is effective for studying walk generating functions.

The approach clarifies the differential algebraic complexity of these combinatorial objects.

Abstract

We use Galois theory of difference equations to study the nature of the generating series of (weighted) walks in the quarter plane with genus zero kernel curve. Using this approach, we prove that the generating series do not satisfy any nontrivial (possibly nonlinear) algebraic differential equation with rational coefficients.