of the weight lattice of $A_2$. The fundamental region $R$ of  [@Koornwinder1-4] coincides with the fundamental region $F(A_2)$ in our notations. The corresponding isometry group is the affine Weyl group of $A_2$.

Furthermore, continuing the comparison with the paper [@Koornwinder1-4], we want to point out that orbit functions are eigenfunctions not only of the Laplace operator written in the appropriate basis, e.g. in $\omega$-basis, the corresponding eigenvalues bring $-4\pi^2 \l\lambda ,\lambda\r$, where $\lambda$ is the representative from the dominant Weyl chamber, which labels the orbit function. This property holds not only for the Lie algebra $A_n$ and its Laplace operator, but also for the differential operators built from the elementary symmetric polynomials, see [@KlimykPatera2006; @KlimykPatera2007-1].

An independent approach to the polynomials in two variables is proposed in [@suetin2], and the generalization of classical Chebyshev polynomials to the case of several variables is also presented in [@EierLidl]. A detailed comparison would be a major task because the results are not explicit and contain no examples of polynomials.

Multivariate exponential functions
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In this section, we consider one more class of special functions, which, as it will be shown, are closely related to orbit functions of $A_n$. Such a relation allows us 