Abstract
Recently, Bostan and his coauthors investigated lattice walks restricted to the non-negative octant $\mathbb{N}^3$. For the $35548$ non-trivial models with at most six steps, they found that many models associated to a group of order at least $200$ and conjectured these groups were in fact infinite groups. In this paper, we first confirm these conjectures and then consider the non-$D$-finite property of the generating function for some of these models.
Highlights
The objective of this paper is to use the properties of Jacobian matrix at fixed points to derive the infiniteness of groups associated with certain lattice walks restricted to the positive octant
We present the non-D-finiteness of corresponding generating functions for some lattice walks of infinite order by considering the asymptotic behavior of their coefficients
In the past few years, lattice path models restricted to the quarter plane and the positive octant have received special attention, and recent works [1,2,3,4,9,11,13] have shown how they can help us better understand generating functions of lattice walks
Summary
The objective of this paper is to use the properties of Jacobian matrix at fixed points to derive the infiniteness of groups associated with certain lattice walks restricted to the positive octant. The 23rd model, known as Gessel walks, was proven D-finite, and even algebraic, by Bostan and Kauers [2] It was conjectured in [4] that the 56 remaining models with infinite group had non-D-finite generating functions. For the cases of models of dimension two, Bostan et al [1] showed that there were 527 models of cardinality at most 6 They found that 118 models are associated to a finite group of order at most 8, and conjectured that the remaining 409 ones associated to a group of infinite order.
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