# Length derivative of the generating series of walks confined in the quarter plane

## TL;DR

This paper investigates the differential algebraic properties of generating functions for quarter-plane walks, revealing conditions under which these functions satisfy algebraic differential relations with respect to length and position variables.

## Contribution

It establishes a precise link between differential relations in length and position variables for generating functions of quarter-plane walks, advancing the understanding of their algebraic nature.

## Key findings

- In the unweighted case, $Q(x,y,t)$ satisfies an algebraic differential relation in $t$ iff in $x$ or $y$
- Characterizes $t$-differential transcendence for 79 walk models
- Uses difference Galois theory to analyze the generating series

## Abstract

In the present paper, we use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane. These series are trivariate formal power series $Q(x,y,t)$ that count the number of walks confined in the first quadrant of the plane with a fixed set of admissible steps, called the model of the walk. While the variables $x$ and $y$ are associated to the ending point of the path, the variable $t$ encodes its length. In this paper, we prove that in the unweighted case, $Q(x,y,t)$ satisfies an algebraic differential relation with respect to $t$ if and only if it satisfies an algebraic differential relation with respect $x$ (resp. $y$). Combined with other papers, we are able to characterize the $t$-differential transcendence of the $79$ models of walks listed by Bousquet-M\'elou and Mishna.