New steps in walks with small steps in the quarter plane

Abstract

In this article we obtain new expressions for the generating functions counting (non-singular) walks with small steps in the quarter plane. Those are given in terms of infinite series, while in the literature, the standard expressions use solutions to boundary value problems. We illustrate our results with three examples (an algebraic case, a transcendental D-finite case, and an infinite group model).

47 pages, 8 figures, to appear in Annals of Combinatorics

References in corpus (2)

• Singularity analysis via the iterated kernel method
• On the Holonomy or Algebraicity of Generating Functions Counting Lattice Walks in the Quarter-Plane

Cited by in corpus (6)

• Differential transcendence & algebraicity criteria for the series counting weighted quadrant walks
• Continued Classification of 3D Lattice Walks in the Positive Octant
• On the nature of four models of symmetric walks avoiding a quadrant
• Quadrant Walks Starting Outside the Quadrant
• Quarter-plane lattice paths with interacting boundaries: the Kreweras and reverse Kreweras models
• On the analysis of partially homogeneous nearest-neighbour random walks in the quarter plane