into the structure of the theory of polynomials. In particular, the generalization of Chebyshev polynomials to any number of variables was a straightforward task. It is based on the Weyl group $W(A_n)$, where $n<\infty$. This only recently became possible, after the orbit functions of simple Lie algebras were introduced as useful special functions [@Patera] and studied in great detail and generality [@KlimykPatera2006; @KlimykPatera2007-1; @KlimykPatera2008].

We proceed in three steps. In Section 2, we exploit the isomorphism of the group of permutations of $n+1$ elements ${\rm S}$ and the Weyl group of ${\rm SU}(n+1)$, or equivalently of $A_n$, and define the orbit functions of $A_n$. This opens the possibility to write the orbit functions in two rather different bases, the orthnormal basis, and the basis determined by the simple roots of $A_n$, which considerably alters the appearance of the orbit functions. In the paper, we use the non-orthogonal basis because of its direct generalization to simple Lie algebras of other types than $A_n$.

In Section 3 we consider classical Chebyshev polynomials of the first and second kind, and compare them with the $C$- and $S$-orbit functions of $A_1$. We show that polynomials of the first kind are in one-to-one correspondence with 