A human proof of Gessel’s lattice path conjecture

A. Bostan,I. Kurkova,K. Raschel

doi:10.1090/tran/6804

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.

Highlights

Gessel walks are lattice paths confined to the quarter plane N2 = {0, 1, 2, . . .} × {0, 1, 2, . . .}, that start at the origin (0, 0) and move by unit steps in one of the following directions: West, East, South-West and North-East, see Figure 1
Gessel walks have been puzzling the combinatorics community since 2001, when Ira Gessel conjectured: (A) For all n 0, the following closed-form expression holds for the number of Gessel excursions of even length 2n q(0, 0; 2n) = 16n (5/6)n(1/2)n, (1)
In this article we have presented the first human proofs of Gessel conjecture (Problem (A)) and of the algebraicity of the complete generating function (GF) counting Gessel walks (Problem (B))

Summary

Introduction

The representation of [22] seems to be hardly accessible for further analyses, such as for expressing the coefficients q(i, j; n) in any satisfactory manner, and in particular for providing a proof of Gessel’s conjecture This approach has been generalized subsequently for all models of walks with small steps in the quarter plane, see [34]. (V) Evaluating the so-obtained expression of Q(x, 0; z) at x = 0 and performing further simplifications (based on several identities involving special functions [1], and on the theory of the Darboux coverings for tetrahedral hypergeometric equations [37]), we shall obtain the solution of Problem (A), and, in this way, the first human proof of Gessel’s conjecture, see Section 4

Meromorphic continuation of the generating functions

Generating functions in terms of Weierstrass zeta functions

Conclusion