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

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-Mélou and Mishna.

References in corpus (2)

• Walks in the quarter plane, genus zero case
• On the Kernel curves associated with walks in the quarter plane

Cited by in corpus (9)

• Intervals in the greedy Tamari posets
• Counting quadrant walks via Tutte's invariant method
• Walks in the quarter plane, genus zero case
• On the Kernel curves associated with walks in the quarter plane
• On the nature of four models of symmetric walks avoiding a quadrant
• On the stationary distribution of reflected Brownian motion in a wedge: differential properties
• Differential algebraic generating series of weighted walks in the quarter plane
• Geometric Invariants of Plane and Space Curves
• Enumeration of three quadrant walks with small steps and walks on other M-quadrant cones