Walks confined in a quadrant are not always D-finite
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.
Cite
@article{arxiv.math/0211432,
title = {Walks confined in a quadrant are not always D-finite},
author = {Mireille Bousquet-Melou and Marko Petkovsek},
journal= {arXiv preprint arXiv:math/0211432},
year = {2025}
}
Comments
To appear in Theoret. Comput. Sci. (special issue devoted to random generation of combinatorial objects and bijective combinatorics)