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There is another approach to multivariate orthogonal polynomials, which is also based on orbit functions. Such polynomials can be constructed by the continuous and invertible change of variables $$\begin{gathered}
\label{subst_exp}
y_j=e^{2\pi i x_j}, \quad x_j\in\R,\quad j=1,2,\dots,n.\end{gathered}$$ Consider an $A_n$ orbit function $C_\lambda(x)$, $S_\lambda(x)$ or $E_\lambda(x)$, when $\lambda$ is given in the $\w$-basis and $x$ is given in the $\alpha$-basis. Each of these functions consists of summands $\prod\limits^n_{j=1}e^{2\pi i \mu_j x_j}$, where $\mu_j\in\Z$ are coordinates of an orbit point $\mu$. Then the summand is transformed by (\[subst\_exp\]) into a monomial of the form $\prod\limits^n_{j=1}y_j^{\mu_j}$. It is convenient to label these polynomials by non-negative integer coordinates $(m_1,m_2,\dots, m_n)$ of the point $\lambda=m_1\w_1+m_2\w_2+\dots m_n\w_n$ and to denote the polynomial obtained from the orbit function $C_\lambda$ as $P_{(m_1,\dots, m_n)}^{C}$ (analogously for $S$ and $E$ functions). Polynomials of two variables obtained from the orbit functions by the substitution (\[subst\_exp\]) are already described in the literature [@Koornwinder1-4], where they are derived from very different considerations. The detailed comparison is made in the following example.

Consider the $A_2$ Weyl orbits of the lower weights $(0,m)_\omega$, $(m,0)_\omega$ and the orbit of the generic point $(m_1,m_2)_\omega$, $m,m_1,m_2\in\Z^{>0}$ $$\begin{gathered}
W_{(0,m)}(A_2)=\{(0, m),\, ( {-}m, 0),\, (m, {-}m)\},
\quad W_{(m,0)}(A_2)=\{(m, 0),\, 