variables, and $C$ orbit functions of $A_n$ $D^+_{\lambda}(x)=kC_\lambda(x)$, where $k=\tfrac{|W|}{|W_\lambda|}$, $|W|$ and $|W_\lambda|$ are sizes of the Weyl group and Weyl orbit respectively. In particular, for generic points, $k=1$.

Proof follows from the definitions of the functions $C$ and $D^+$ (definitions 1 and 4 respectively) and properties of orbit functions formulated in Section \[orb\_func\_n+1\].

[@KlimykPatera2007-3] For a fixed point $\lambda=(l_1,l_2,\dots,l_{n+1})_e$, such that $l_1\geq l_2\geq\cdots\geq l_{n+1}$, ${\sum\limits_{k=1}^{n+1}l_k=0}$, the antisymmetric multivariate exponential function $D^-_{\lambda}$ of $x=(x_1,x_2,\dots,x_{n+1})_e\in \mathcal{H}$ is defined as follows $$\begin{gathered}
\label{E+def}
D^-_{\lambda}(x):=\det \left(
\begin{array}{cccc}
e^{2\pi i l_1x_1}&e^{2\pi i l_1x_2}&\dots&e^{2\pi i l_1x_{n+1}}\\
e^{2\pi i l_2x_1}&e^{2\pi i l_2x_2}&\dots&e^{2\pi i l_2x_{n+1}}\\[-1ex]
\vdots&\vdots&\ddots&\vdots\\[1 ex]
e^{2\pi i l_{n+1}x_1}&e^{2\pi i l_{n+1}x_2}&\dots&e^{2\pi i l_{n+1}x_{n+1}}
\end{array}
\right)=\sum_{s\in {\rm S}_{n+1}}({\rm sgn}\ s)e^{2\pi i(s(\lambda),x)},\end{gathered}$$ where ${\rm sgn}$ is the permutation sign.

For all generic points $\lambda\in \mathcal{H}\subset\R^{n+1}$, we have the following connection ${D^-_{\lambda}(x)=S_\lambda(x)}$.

The antisymmetric multivariate exponential functions $D^-$, and $S$ orbit functions, equal zero for non-generic points.

Proof directly follows from the definitions of functions $S$ and $D^-$ (definitions 2 and 5 respectively), and properties of $S$ functions formulated in Section \[orb\_func\_n+1\].

 [@KlimykPateraAlternatingExp] The alternating multivariate exponential function $D^{{\rm Alt}}_\lambda(x)$, for $x=(x_1,\dots,x_{n+1})_e$, $\lambda=(l_1,\dots,l_{n+1})_e$, is defined as the function $$\begin{gathered}
\label{sdetD_Alt}
D^{{\rm Alt}}_\lambda(x):={\rm sdet}\left(
\begin{array}{cccc}
e^{2\pi i l_1x_1}    & e^{2\pi 