Differential algebraic generating series of weighted walks in the quarter plane

TL;DR

This paper investigates the algebraic differential equations satisfied by the generating series of weighted walks in the quarter plane, establishing conditions under which these series satisfy such equations in multiple variables.

Contribution

It proves that the generating series satisfies a nontrivial algebraic differential equation in one variable if and only if it does so in all variables, completing previous results.

Findings

Series satisfies algebraic differential equations in all variables or none

Provides conditions linking single-variable and multivariable differential equations

Completes the proof of a key theorem in weighted walks analysis

Abstract

In the present paper we study the nature of the trivariate generating series of weighted walks in the quarter plane. Combining the results of this paper to previous ones, we complete the proof of the following theorem. The series satisfies a nontrivial algebraic differential equation in one of its variable, if and only if it satisfies a nontrivial algebraic differential equation in each of its variables.