original result allows for asymptotic expansions of coefficient sequences more generally defined from multivariate functions than the diagonal sequence).

\[thm:expGen\] Suppose that the meromorphic function $F({\ensuremath{\mathbf{z}}},t) = G({\ensuremath{\mathbf{z}}},t)/H({\ensuremath{\mathbf{z}}},t)$ has an isolated strictly minimal simple pole at $({\ensuremath{\mathbf{w}}},t_{\ensuremath{\mathbf{w}}})$. If $tH_t$ does not vanish at $({\ensuremath{\mathbf{w}}},t_{\ensuremath{\mathbf{w}}})$ then there is an asymptotic expansion $$c_n \sim (w_1\cdots w_d \cdot t)^{-n} \sum_{l \geq l_0} C_l n^{-(d+l)/2} \label{eq:Genasm}$$ for constants $C_l$, where $l_0$ is the degree to which $G$ vanishes near $({\ensuremath{\mathbf{w}}},t_{\ensuremath{\mathbf{w}}})$. When $G$ does not vanish at $({\ensuremath{\mathbf{w}}},t_{\ensuremath{\mathbf{w}}})$ then $l_0 = 0$ and the leading term of this expansion is $$C_0 = (2\pi)^{-d/2}\mathcal{H}_{{\ensuremath{\mathbf{w}}}}^{-1/2} \cdot \frac{G({\ensuremath{\mathbf{w}}},t_{\ensuremath{\mathbf{w}}})}{t H_t({\ensuremath{\mathbf{w}}},t_{\ensuremath{\mathbf{w}}})}.  \label{eq:domterm}$$

In fact, Corollary 3.7 of [@PeWi02] shows that in the case of a finitely minimal point one can simply sum the contributions of each point. Combining this with the above calculations gives our main result.

\[thm:asm\] Let ${\ensuremath{\mathcal{S}}}\subseteq \{-1,0,1\}^d \setminus \{\mathbf{0}\}$ be a set of unit steps in dimension $d$. If ${\ensuremath{\mathcal{S}}}$ is symmetric with respect to each axis, and ${\ensuremath{\mathcal{S}}}$ takes a positive step in each direction, then the number of walks of length $n$ taking steps in ${\ensuremath{\mathcal{S}}}$, beginning at the origin, and never leaving the positive orthant has asymptotic expansion $$\label{eq:thmexp}
s_n  = \left[ \left(s^{(1)} 