$C$-functions. Polynomials of the second kind coincide with the appropriate $S$-function divided by the unique lowest non-trivial $S$-function. We point out that polynomials of the second kind can be identified as irreducible characters of finite dimensional representations of ${\rm SU}(2)$. Useful properties of Chebyshev polynomials can undoubtedly be traced to that identification, because the fundamental object of representation theory of semisimple Lie groups/algebras is character. In principle, all one needs to know about an irreducible finite dimensional representation can be deduced from its character. An important aspect of this conclusion is that characters are known and uniformly described for all simple Lie groups/algebras.

In Section 4 we provide details of the recursive procedure from which the analog of the trigonometric form of Chebyshev polynomials in $n$ variables can be found. Thus there are $n$ generic recursion relations for $A_n$, having at least $n+2$ terms, and at most $\left(\begin{smallmatrix}n+1\\ [(n+1)/2]\\\end{smallmatrix}\right)+1$ terms. Irreducible polynomials are divided into $n+1$ exclusive classes with the property that monomials within one irreducible polynomial belong to the same class. This follows directly from the recognition of the presence and properties of the underlying Lie algebra.

In subsection 4.2, the simple substitution $z=e^{2\pi ix}$, $x\in\R^n$, is used in 