On the nature of four models of symmetric walks avoiding a quadrant

TL;DR

This paper investigates the generating series of four symmetric walk models in the quarter plane, revealing that three are differentially transcendental while one is differentially algebraic, using advanced algebraic methods.

Contribution

It characterizes the differential nature of generating series for four symmetric walk models with infinite groups and genus-one kernels, identifying one as differentially algebraic.

Findings

Three models have differentially transcendental generating series.

One model has a differentially algebraic generating series.

The analysis uses difference Galois theory to classify the series.

Abstract

We study the nature of the generating series of some models of walks with small steps in the three quarter plane. More precisely, we restrict ourselves to the situation where the group is infinite, the kernel has genus one, and the step set is diagonally symmetric (i.e., with no steps in anti-diagonal directions). In that situation, after a transformation of the plane, we derive a quadrant-like functional equation. Among the four models of walks, we obtain, using difference Galois theory, that three of them have a differentially transcendental generating series, and one has a differentially algebraic generating series.