to view orbit functions in the orthonormal basis, and to represent them in the form of determinants and permanents. At the same time, we obtain the straightforward procedure for constructing polynomials from multivariate exponential functions.

[@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 symmetric multivariate exponential function $D^+_{\lambda}$ of $x=(x_1,x_2,\dots,x_{n+1})_e$ is defined as follows $$\begin{gathered}
\label{E+def}
D^+_{\lambda}(x):= {\det}^+ \left(
\begin{array}{cccc}
e^{2\pi i l_1x_1}&e^{2\pi i l_1x_2}&\dots&e^{2\pi i l_1x_{n+1}}\\
e^{2\pi i l_2x_1}&e^{2\pi i l_2x_2}&\dots&e^{2\pi i l_2x_{n+1}}\\[-1ex]
\vdots&\vdots&\ddots&\vdots\\[1 ex]
e^{2\pi i l_{n+1}x_1}&e^{2\pi i l_{n+1}x_2}&\dots&e^{2\pi i l_{n+1}x_{n+1}}
\end{array}
\right).\end{gathered}$$ Here, ${\det}^+$ is calculated as a conventional determinant, except that all of its monomial terms are taken with positive sign. It is also called *permanent* [@Henryk] or *antideterminant*.

It was shown in [@KlimykPatera2007-3] that it suffices to consider $D_{\lambda}^+(x)$ on the hyperplane $x\in {\mathcal H}$ (see (\[plane H\])). Furthermore, due to the following property of the permanent $$\begin{gathered}
{\det}^+ (a_{ij})_{i,j=1}^m=\sum_{s\in {\rm S}_m}a_{1,s(1)}a_{2,s(2)}\cdots a_{m,s(m)}
=\sum_{s\in {\rm S}_m}a_{s(1),1}a_{s(2),2}\cdots a_{s(m),m}\end{gathered}$$ we have $$\begin{gathered}
D^+_{\lambda}(x)= \sum_{s\in {\rm S}_{n+1}}e^{2\pi i l_1 x_{s(1)}} \cdots
e^{2\pi i l_m x_{s(n+1)}} =\sum_{s\in {\rm S}_{n+1}}e^{2\pi i
(\lambda,s(x))} =\sum_{s\in {\rm S}_{n+1}}e^{2\pi i (s(\lambda),x)}.\end{gathered}$$

For all $\lambda, x\in \mathcal{H}\subset\R^{n+1}$, we have the following connection between the symmetric multivariate exponential functions in $n+1$ 