W_\lambda} (-1)^{p(\mu)}e^{2\pi i \sum\limits^n_{j=1}\mu_jx_j}
= \sum_{\mu\in W_\lambda} (-1)^{p(\mu)}\prod\limits^n_{j=1} e^{2\pi i \mu_jx_j},\label{def_s-function2}
\\
E_\lambda(x)
= \sum_{\mu\in W^e_{\lambda}} e^{2\pi i \sum\limits^n_{j=1}\mu_jx_j}
= \sum_{\mu\in W^e_{\lambda}} \prod\limits^n_{j=1} e^{2\pi i \mu_jx_j}.\label{def_e-function2}\end{gathered}$$

Some properties of orbit functions
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For $S$ functions, the number of summands is always equal to the size of the Weyl group. Note that in the 1-dimensional case, $C$-, $S$- and $E$-functions are respectively a cosine, a sine and an exponential functions up to the constant.

All three families of orbit functions are based on semisimple Lie algebras. The number of variables coincides with the rank of the Lie algebra. In general, $C$-, $S$- and $E$- functions are finite sums of exponential functions. Therefore they are continuous and have continuous derivatives of all orders in $\R^n$.

The $S$-functions are antisymmetric with respect to the $(n{-}1)$-dimensional boundary of $F$. Hence they are zero on the boundary of $F$. The $C$-functions are symmetric with respect to the $(n-1)$-dimensional boundary of $F$. Their normal derivative at the boundary is equal to zero (because the normal derivative of a $C$-function is an $S$-function).

For simple Lie algebras of any type, the functions $C_\lambda(x)$, $E_\lambda(x)$ and $S_\lambda(x)$ are eigenfunctions of the appropriate Laplace operator. The Laplace operator has 