Gessel and Zeilberger generalized the reflection principle to handle walks confined to Weyl chambers, under some restrictions on the allowable steps. For those models that are invariant under the Weyl group action, they express the counting function for the walks with fixed starting and endpoint as a constant term in the Taylor series expansion of a rational function. Here, we focus on the simplest case, the Weyl groups $A_1^d$, which correspond to walks in the first orthant $\mathbb{N}^d$ taking steps from a subset of $\{\pm1, 0\}^d$ which is invariant under reflection across any axis. The principle novelty here is the incorporation of weights on the steps and the main result is a very general theorem giving asymptotic enumeration formulas for walks that end anywhere in the orthant. The formulas are determined by singularity analysis of multivariable rational functions, an approach that has already been successfully applied in numerous related cases.

中文翻译：

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任意维度可反射加权游走的渐近性

Gessel 和 Zeilberger 将反射原理概括为处理限于 Weyl 室的步行，但对允许的步骤有一些限制。对于那些在外尔群作用下不变的模型，它们将具有固定起点和终点的步行的计数函数表示为有理函数的泰勒级数展开式中的常数项。在这里，我们关注最简单的情况，Weyl 群 $A_1​​^d$，它对应于从 $\{\pm1, 0\} 的子集走步的第一个 orthant $\mathbb{N}^d$ ^d$ 在任何轴的反射下都是不变的。这里的原则新颖性是在步骤上加入权重，主要结果是一个非常普遍的定理，给出了在 orthant 中任何地方结束的步行的渐近枚举公式。