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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The complete generating function for Gessel walks is algebraic
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by Alin Bostan and Manuel Kauers; \\with an appendix by Mark van Hoeij PDF
Proc. Amer. Math. Soc. 138 (2010), 3063-3078 Request permission


Gessel walks are lattice walks in the quarter-plane $\mathbb N^2$ which start at the origin $(0,0)\in \mathbb N^2$ and consist only of steps chosen from the set $\{\leftarrow , \swarrow , \nearrow , \rightarrow \}$. We prove that if $g(n;i,j)$ denotes the number of Gessel walks of length $n$ which end at the point $(i,j)\in \mathbb N^2$, then the trivariate generating series $\displaystyle {G(t;x,y)=\sum _{n,i,j\geq 0} g(n;i,j)x^i y^j t^n}$ is an algebraic function.
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Additional Information
  • Alin Bostan
  • Affiliation: Algorithms Project, INRIA Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt-B.P. 105, 78153 Le Chesnay Cedex, France
  • MR Author ID: 725685
  • Email:
  • Manuel Kauers
  • Affiliation: RISC, Johannes Kepler University, Altenbergerstrasse 69, A-4040 Linz, Austria
  • Email:
  • Mark van Hoeij
  • Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306
  • Email:
  • Received by editor(s): September 26, 2009
  • Published electronically: May 14, 2010
  • Additional Notes: The first author was partially supported by the Microsoft Research-INRIA Joint Centre
    The second author was partially supported by Austrian Science Fund (FWF) grant no. P19462-N18.
  • Communicated by: Jim Haglund
  • © Copyright 2010 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 3063-3078
  • MSC (2010): Primary 05A15, 14N10, 33F10, 68W30; Secondary 33C05, 97N80
  • DOI:
  • MathSciNet review: 2653931