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Journal Articles Transactions of the American Mathematical Society Year : 2017

A human proof of Gessel's lattice path conjecture

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.
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Dates and versions

hal-00858083 , version 1 (04-09-2013)
hal-00858083 , version 2 (13-02-2014)
hal-00858083 , version 3 (13-02-2015)

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Alin Bostan, Irina Kurkova, Kilian Raschel. A human proof of Gessel's lattice path conjecture. Transactions of the American Mathematical Society, 2017, 369 (2, February 2017), pp.1365-1393. ⟨hal-00858083v3⟩
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