hull of the vertices $\{0,\omega_1,\omega_2,\ldots, \omega_n\}$.

The $C$ orbit function $C_{\lambda}(x)$, $\lambda\in P^+$ is defined as $$\begin{gathered}
\label{def_c-function1}
C_\lambda(x) := \sum_{\mu\in W_\lambda(G)} e^{2\pi i \l\mu, x\r},
\qquad
x\in\R^n.\end{gathered}$$

The $S$ orbit function $S_{\lambda}(x)$, $\lambda\in P^{++}$ is defined as $$\begin{gathered}
\label{def_s-function1}
S_\lambda(x) := \sum_{\mu\in W_\lambda(G)} (-1)^{p(\mu)}e^{2\pi i \l\mu , x\r},\qquad
x\in\R^n,\end{gathered}$$ where $p(\mu)$ is the number of reflections necessary to obtain $\mu$ from $\lambda$. Of course the same $\mu$ can be obtained by different successions of reflections, but all routes from $\lambda$ to $\mu$ will have a length of the same parity, and thus the salient detail given by $p(\mu)$, in the context of an $S$-function, is meaningful and unchanging.

We define $E$ orbit function $E_{\lambda}(x)$, $\lambda\in P^e$ as $$\begin{gathered}
\label{def_e-function1}
E_\lambda(x) := \sum_{\mu\in W^e_{\lambda}(G)} e^{2\pi i \l\mu, x\r},
\qquad
x\in\R^n,\end{gathered}$$ where $P^e:=P^+\cup r_i P^+$ and $r_i$ is a reflection from $W$.

If we always suppose that $\lambda, \mu\in P$ are given in the $\w$-basis, and $x\in\R^n$ is given in the $\alpha$ basis, namely $\lambda=\sum\limits^n_{j=1}\lambda_j\w_j$, $\mu=\sum\limits^n_{j=1}\mu_j\w_j$, $\lambda_j, \mu_j\in\Z$ and $x=\sum\limits^n_{j=1}x_j\alpha_j$, $x_j\in \R$, then the orbit functions of $A_n$ have the following forms $$\begin{gathered}
C_\lambda(x)
= \sum_{\mu\in W_\lambda} e^{2\pi i \sum\limits^n_{j=1}\mu_jx_j}
= \sum_{\mu\in W_\lambda} \prod\limits^n_{j=1} e^{2\pi i \mu_jx_j},\label{def_c-function2}
\\
S_\lambda(x)
= \sum_{\mu\in 