studied in the context of Gessel and Zeilberger [@GeZe92].

### A telescoping sum

Next we apply each of the $2^d$ elements of $\mathcal{G}$ to Equation , and take a weighted sum. Define ${\operatorname{sgn}}(\sigma)=(-1)^r$, where $r = \# \{k : \sigma(z_k)={\ensuremath{\overline{z}}}_k\}$, and let $\sigma_k$ be the map which sends $z_k$ to ${\ensuremath{\overline{z}}}_k$ and fixes all other components of $(z_1,\dots,z_d)$.

Let $F({\ensuremath{\mathbf{z}}},t)$ be the generating function counting the number of walks of length $n$ with marked endpoint. Then, as elements of the ring $\mathbb{Q}[z_1,{\ensuremath{\overline{z}}}_1,\dots,z_d,{\ensuremath{\overline{z}}}_d][\![t]\!]$, $$\sum_{\sigma \in \mathcal{G}}{\operatorname{sgn}}(\sigma)\cdot \sigma(z_1\cdots z_d) \sigma(F({\ensuremath{\mathbf{z}}},t)) = \frac{\sum_{\sigma \in \mathcal{G}}{\operatorname{sgn}}(\sigma)\cdot \sigma(z_1\cdots z_d)}{1-tS({\ensuremath{\mathbf{z}}}) }. \label{eq:orbit}$$

For each $\sigma \in \mathcal{G}$ we have ${\operatorname{sgn}}(\sigma) =
-{\operatorname{sgn}}(\sigma_k\sigma)$ and, for the $A_k$ in Equation , $$\sigma(A_k(z_1,\dots,z_{k-1},z_{k+1}\dots,z_d,t)) =
  (\sigma_k\sigma)(A_k(z_1,\dots,z_{k-1},z_{k+1}\dots,z_d,t)).$$ Thus, we can apply each $\sigma \in \mathcal{G}$ to Equation and sum the results, weighted by ${\operatorname{sgn}}(\sigma)$, to cancel each $A_k$ term on the right hand side. Minor algebraic manipulations, along with the fact that the group elements fix $S(z_1,\dots,z_d)$, then give Equation .

### Positive series extraction

Next, we note that each term in the expansion of $$\sigma_1(z_1,\dots,z_d) \sigma_1(F({\ensuremath{\mathbf{z}}},t)) = - ({\ensuremath{\overline{z}}}_1z_2\cdots z_d) F({\ensuremath{\overline{z}}}_1,z_2,\dots,z_d,t) \in \mathbb{Q}[z_1,{\ensuremath{\overline{z}}}_1,\dots,z_d,{\ensuremath{\overline{z}}}_d][\![t]\!]$$ has a negative power of $z_1$. In fact, except for when $\sigma$ is the identity any summand 