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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2002
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/math/0211432 |
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| _version_ | 1866909806051196928 |
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| author | Bousquet-Melou, Mireille Petkovsek, Marko |
| author_facet | Bousquet-Melou, Mireille Petkovsek, Marko |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0211432 |
| institution | arXiv |
| publishDate | 2002 |
| record_format | arxiv |
| spellingShingle | Walks confined in a quadrant are not always D-finite Bousquet-Melou, Mireille Petkovsek, Marko Combinatorics 05A15 (primary) 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. |
| title | Walks confined in a quadrant are not always D-finite |
| topic | Combinatorics 05A15 (primary) |
| url | https://arxiv.org/abs/math/0211432 |