Explicit expression for the generating function counting Gessel's walks

Abstract

Gessel's walks are the planar walks that move within the positive quadrant ${\mathbb{Z}}_{+}^2$ by unit steps in any of the following directions: West, North-East, East and South-West. In this paper, we find an explicit expression for the trivariate generating function counting the Gessel's walks with $k\ge 0$ steps, which start at $(0,0)$ and end at a given point $(i,j)\in {\mathbb{Z}}_{+}^2$.

23 pages

References in corpus (3)

• Walks with small steps in the quarter plane
• Proof of Ira Gessel's Lattice Path Conjecture
• Towards a human proof of Gessel's conjecture