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arXiv · math/0211432

Walks confined in a quadrant are not always D-finite

Mireille Bousquet-Melou·Marko Petkovsek
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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 Z^2, 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.

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Mireille Bousquet-Melou, Marko Petkovsek. 2002-11-27. Walks confined in a quadrant are not always D-finite. https://doi.org/10.1016/s0304-3975(03)00219-6

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