# On the nature of the generating series of walks in the quarter plane

## TL;DR

This paper introduces a novel Galois-theoretic approach to analyze the generating series of walks in the quarter plane, establishing new hypertranscendency results that show these series do not satisfy certain algebraic differential equations.

## Contribution

It applies Galois theory of difference equations to study these generating series, providing new hypertranscendency results beyond previous findings.

## Key findings

- Recovered recent results on generating series
- Proved certain series are hypertranscendental
- Established nonexistence of specific algebraic differential equations

## Abstract

In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many of the recent results about these series, but also to go beyond them. For instance, we give for the first time hypertranscendency results, {\it i.e.}, we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational coefficients.