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Walks confined in a quadrant are not always D-finite. (English) Zbl 1070.68112
Summary: We consider planar lattice walks that start from a prescribed position, take their steps in a given finite subset of $$\mathbb Z^2$$ , and always stay in the quadrant $$x\geqslant 0$$, $$y\geqslant 0$$. 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.

MSC:
 68R05 Combinatorics in computer science
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References:
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