functions of $A_1$ and their polynomial form {#recursive_polynomials}

In this subsection, we start a derivation of the $A_1$ polynomials in a way which emphasizes the role of the Lie algebra and, more importantly, in a way that directly generalizes to simple Lie algebras of any rank $n$ and any type, resulting in polynomials of $n$ variables and of a new type for each algebra. In the present case of $A_1$, this leads us to a different normalization of the polynomials and their trigonometric variables than is common for classical Chebyshev polynomials. No new polynomials emerge than those equivalent to Chebyshev polynomials of the first and second kind. Insight is nevertheless gained into the structure of the problem, which, to us, turned out to be of considerable importance. We are inclined to consider the Chebyshev polynomials, in the form derived here, as the canonical polynomials.

The underlying Lie algebra $A_1$ is often denoted $sl(2,\C)$ or $su(2)$. In fact, this case is so simple that the presence of the Lie algebras has never been acknowledged.

The orbit functions of $A_1$ are of two types (\[c-func\_A1\]) and (\[s-func\_A1\]); in particular, $C_0(x)=2$, and $S_0(x)=0$ for all $x$.

The simplest substitution of variables to transform 