Walks confined in a quadrant are not always D-finite

  • Authors:

  • Mireille Bousquet-Mélou;Marko Petkovšek

  • Affiliations:

  • CNRS, LaBRI, Université Bordeaux 1, 351 cours de la Libération, F-33405 Talence, Cedex, France;Department of Mathematics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia

  • Venue:
  • Theoretical Computer Science - Random generation of combinatorial objects and bijective combinatorics
  • Year:
  • 2003

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Abstract

We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of Z2, and always stay in the quadrant x ≥ 0, y ≥ 0. We first give a criterion which guarantees that the length generating function of these walks is D-finite, that is, satisfies a linear differential equation with polynomial coefficients. This criterion applies, among others, to the ordinary square lattice walks. Then, we prove that walks that start from (1,1), take their steps in {(2,-1), (-1, 2)} and stay in the first quadrant have a non-D-finite generating function. Our proof relies on a functional equation satisfied by this generating function, and on elementary complex analysis.