The generating function of Kreweras walks with interacting boundaries is not algebraic

TL;DR

This paper investigates the algebraic nature of generating functions for Kreweras walks with interacting boundaries, confirming that they are generally not algebraic except in specific symmetric cases.

Contribution

The paper uses computer algebra tools to demonstrate that the generating function for Kreweras walks with interactions is not algebraic in general, resolving an open question.

Findings

Generating function is not algebraic except in symmetric interaction cases.

Confirms previous conjectures about the non-algebraicity of Kreweras walks.

Provides computational evidence supporting the non-algebraic nature.

Abstract

Beaton, Owczarek and Xu (2019) studied generating functions of Kreweras walks and of reverse Kreweras walks in the quarter plane, with interacting boundaries. They proved that for the reverse Kreweras step set, the generating function is always algebraic, and for the Kreweras step set, the generating function is always D-finite. However, apart from the particular case where the interactions are symmetric in $x$ and~$y$, they left open the question of whether the latter one is algebraic. Using computer algebra tools, we confirm their intuition that the generating function of Kreweras walks is not algebraic, apart from the particular case already identified.