orbit functions. For some applications/calculations it is convenient to scale orbit functions of non-generic points on the factor equal to the order of the stabilizer of that point in the Weyl group $W(A_1)$.

Orbit functions of $A_n$ and their polynomials
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This section proposes two approaches to constructing orthogonal polynomials of $n$ variables based on orbit functions. The first comes from the decomposition of Weyl orbit products into sums of orbits. Its result is the analog of the trigonometric form of the Chebyshev polynomials. The second approach is the exponential substitution in [@KlimykPatera2006].

Recursive construction
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Since the $C$- and $S$- functions are defined for $A_n$ of any rank $n=1,2,3,\dots$, it is natural to take $C$-functions and the ratio of $S$-functions as multidimensional generalizations of Chebyshev polynomials of the first and second kinds respectively $$\begin{aligned}
T_\lambda(x)&:=C_\lambda(x),\qquad \ x\in\R^n,\\
U_\lambda(x)&:=\tfrac{S_{\lambda+\rho}(x)}{S_{\rho}(x)},\qquad
\rho=\omega_1{+}\omega_2{+}\dots{+}\omega_n=(1,1,\dots,1)_\omega,\quad x\in\R^n,\end{aligned}$$ where $\lambda$ is one of the dominant weights of $A_n$.

The functions $T_\lambda$ and $U_\lambda$ can be constructed as polynomials using the recursive scheme proposed in Section \[recursive\_polynomials\]. In the $n$-dimensional case of orbit functions of $A_n$, we start from the $n$ orbit functions labeled by the fundamental weights, $$\begin{gathered}
X_1:=C_{\omega_1}(x),\quad X_2:=C_{\omega_2}(x),\quad\dots,\quad
X_n:=C_{\omega_n}(x)\,,
        \qquad x\in\R^n\,.\end{gathered}$$ By multiplying them and 