decomposing the products into the sum of orbit functions, we build the polynomials for any $C$- and $S$-function.

The generic recursion relations are found as the decomposition of the products $X_{\w_j}C_{(a_1,a_2.\dots,a_n)}$ with ‘sufficiently large’ $a_1,a_2,\dots,a_n$. Such a recursion relation has $\left(\begin{smallmatrix}n+1\\ j\\\end{smallmatrix}\right)+1$ terms, where $\left(\begin{smallmatrix}n+1\\j\\\end{smallmatrix}\right)$ is the size of the orbit of $\w_j$.

An efficient way to find the decompositions is to work with products of Weyl group orbits, rather than with orbit functions. Their decomposition has been studied, and many examples have been described in [@HLP]. It is useful to be aware of the congruence class of each product, because all of the orbits in its decomposition necessarily belong to that class. The *congruence number* $\#$ of an orbit $\lambda$ of $A_n$, which is also the congruence number of the orbit functions $C_\lambda$ and $S_\lambda$, specifies the class. It is calculated as follows, $$\begin{gathered}
\#(C_{(a_1,a_2,\dots,a_n)}(x))=\#(S_{(a_1,a_2,\dots,a_n)}(x))=\sum_{k=1}^n ka_k\mod (n+1).\end{gathered}$$ In particular, each $X_j$, where $j=1,2,\dots,n$, is in its own congruence class. During the multiplication, congruence numbers add up $\mod n+1$.

Polynomials in two and three variables originating from orbit functions of the simple Lie algebras $A_2$, $C_2$, $G_2$, $A_3$, $B_3$, and $C_3$ are obtained in the forthcoming paper [@NesterenkoPatera2009].

Exponential substitution
