generic points that ${E}_\lambda(x)=\tfrac12\Big({C}_\lambda(x)+{S}_\lambda(x)\Big)$, and from the relations (\[ri\_symm\_of\_C\_S\]), we obtain $$\begin{gathered}
\label{ri_symm_of_E}
{E}_{r_i\lambda}(x)={E}_\lambda(r_ix)=\tfrac12\left( {C}_\lambda(x)-{S}_\lambda(x)\right)={E}_{\lambda}(x).\end{gathered}$$

A number of other properties of orbit functions are presented in [@KlimykPatera2006; @KlimykPatera2007-1; @KlimykPatera2008].

Orbit functions and Chebyshev polynomials {#sec_A1_and_Chebyshev}
=========================================

We recall known properties of Chebyshev polynomials [@Rivlin1974] in order to be subsequently able to make an unambiguous comparison between them and the appropriate orbit functions.

Classical Chebyshev polynomials {#ssec_Classical_Chebyshev}
-------------------------------

Chebyshev polynomials are orthogonal polynomials which are usually defined recursively. One distinguishes between Chebyshev polynomials of the first kind $T_n$: $$\begin{gathered}
T_0(x)=1,\quad
T_1(x)=x,\quad
T_{n+1}(x)=2xT_n-T_{n-1},\label{def_T0_T1_Tn}\\
\text{hence}\quad
T_2(x)=2x^2-1,\quad
T_3(x)=4x^3-3x, \dots\label{def_T2_T3}\end{gathered}$$ and Chebyshev polynomials of the second kind $U_n$: $$\begin{gathered}
U_0(x)=1,\quad
U_1(x)=2x,\quad
U_{n+1}(x)=2xU_n-U_{n-1},\label{def_U0_U1_Un}\\
\text{in particular}\quad
U_2(x)=4x^2-1,\quad
U_3(x)=8x^3-4x, \quad etc.\label{def_U2_U3}\end{gathered}$$

The polynomials $T_n$ and $U_n$ are of degree $n$ in the variable $x$. All terms in a polynomial have the parity of $n$. The coefficient of the leading term of $T_n$ is $2^{n-1}$ and $2^n$ for $U_n$, $n=1,2,3,\dots$.

The roots of the Chebyshev polynomials of the first kind are widely used as nodes for polynomial interpolation in approximation theory. The Chebyshev polynomials are a special case of Jacobi polynomials. They are orthogonal with the following weight functions: $$\begin{gathered}
\int\limits^1_{-1}\frac{1}{\sqrt{1-x^2}}T_n(x)T_m(x){\rm d}x=
\left\{
\begin{array}{l}
0,\quad n\ne m,\\
\pi,\quad n=m=0,\\
