On the D-finiteness of generating functions counting small steps walks in the quadrant
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| Format: | Preprint |
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2025
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| _version_ | 1866914058255466496 |
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| author | Hardouin, Charlotte |
| author_facet | Hardouin, Charlotte |
| contents | The enumeration of small steps walks confined to the first quadrant of the plane has attracted a lot of attention over the past fifteen years. The associated generating functions are trivariate formal power series in $x,y,t$ where the parameter $t$ encodes the length of the walk while the variables $x,y$ correspond to the coordinates of its ending point. These functions satisfy a functional equation in two catalytic variables.
Bousquet-Mélou and Mishna have associated to any small steps model an algebraic curve called the kernel curve and a group called the group of the walk. These two objects turned out to be central in the classification of small steps models. In a recent work, Dreyfus, Elvey Price, and Raschel prove that the group of the walk is finite if and only if the generating function is $D$-finite, that is, it satisfies a linear differential equation with polynomial coefficients in each of its variables $x,y,t$.
In this paper, we show that if the group of the walk is infinite, the generating function doesn't satisfy a linear differential equation in $x,y$ or $t$ over the field $\mathbb{Q}(x,y,t)$. The proof of Dreyfus, Elvey Price, and Raschel is based on some singularity analysis. Here, we propose a new strategy which relies essentially on the aforementioned functional equation and on algebraic arguments. This point of view sheds also a new light on the algebraic nature of the generating functions of small steps models since it relates their $D$-finiteness more directly to some geometric properties of the kernel curve. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_22464 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the D-finiteness of generating functions counting small steps walks in the quadrant Hardouin, Charlotte Combinatorics 05A15, 39A13 The enumeration of small steps walks confined to the first quadrant of the plane has attracted a lot of attention over the past fifteen years. The associated generating functions are trivariate formal power series in $x,y,t$ where the parameter $t$ encodes the length of the walk while the variables $x,y$ correspond to the coordinates of its ending point. These functions satisfy a functional equation in two catalytic variables. Bousquet-Mélou and Mishna have associated to any small steps model an algebraic curve called the kernel curve and a group called the group of the walk. These two objects turned out to be central in the classification of small steps models. In a recent work, Dreyfus, Elvey Price, and Raschel prove that the group of the walk is finite if and only if the generating function is $D$-finite, that is, it satisfies a linear differential equation with polynomial coefficients in each of its variables $x,y,t$. In this paper, we show that if the group of the walk is infinite, the generating function doesn't satisfy a linear differential equation in $x,y$ or $t$ over the field $\mathbb{Q}(x,y,t)$. The proof of Dreyfus, Elvey Price, and Raschel is based on some singularity analysis. Here, we propose a new strategy which relies essentially on the aforementioned functional equation and on algebraic arguments. This point of view sheds also a new light on the algebraic nature of the generating functions of small steps models since it relates their $D$-finiteness more directly to some geometric properties of the kernel curve. |
| title | On the D-finiteness of generating functions counting small steps walks in the quadrant |
| topic | Combinatorics 05A15, 39A13 |
| url | https://arxiv.org/abs/2509.22464 |