  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.
