A human proof of Gessel's lattice path conjecture

TL;DR

This paper provides the first human-derived proofs for Gessel's lattice path conjecture and the algebraic nature of the generating function, using a novel approach involving Weierstrass zeta functions.

Contribution

It introduces a new expression for Gessel walks' generating function in terms of Weierstrass zeta functions, enabling human proofs of previous computer-aided results.

Findings

Confirmed Gessel's conjecture with a human proof.

Established the algebraic nature of the generating function.

Connected Gessel walks to Weierstrass zeta functions.

Abstract

Gessel walks are lattice paths confined to the quarter plane that start at the origin and consist of unit steps going either West, East, South-West or North-East. In 2001, Ira Gessel conjectured a nice closed-form expression for the number of Gessel walks ending at the origin. In 2008, Kauers, Koutschan and Zeilberger gave a computer-aided proof of this conjecture. The same year, Bostan and Kauers showed, again using computer algebra tools, that the complete generating function of Gessel walks is algebraic. In this article we propose the first "human proofs" of these results. They are derived from a new expression for the generating function of Gessel walks in terms of Weierstrass zeta functions.