$\lambda=m\omega_1$ has two points for $m\ne 0$, namely $W_\lambda=\{(m),(-m)\}$. The orbit of $\lambda=0$ has just one point, $W_0=\{0\}$.

One-dimensional orbit functions have the form (see (\[def\_c-function2\]), (\[def\_s-function2\]), (\[def\_e-function2\])) $$\begin{gathered}
C_\lambda(x)=e^{2\pi i m x}{+}e^{{-}2\pi i m x}{=}2\cos(2\pi m x){=}2\cos(m y),
\quad \text{where}\quad y=2\pi x,\; m\in\Z^{\geqslant 0};\label{c-func_A1}
\\
S_\lambda(x)=e^{2\pi i m x}{-}e^{-2\pi i m x}{=}2i\sin(2\pi m x){=}2i\sin(m y),
\quad \text{for}\quad m\in\Z^{> 0};\label{s-func_A1}
\\
E_\lambda(x)= e^{2\pi i m x}=y^m,
\quad \text{where}\quad y=e^{2\pi i x}, \; m\in\Z.\label{e-func_A1}\end{gathered}$$ From (\[c-func\_A1\]) and (\[trigon\_T\]) it directly follows that polynomials generated from $C_m$ functions of $A_1$ are doubled Chebyshev polynomials $T_m$ of the first kind for $m=0,1,2,\dots$.

Analogously, from (\[s-func\_A1\]) and (\[trigon\_U\]), it follows that polynomials $\frac{S_{m+1}}{S_1}$ are Chebyshev polynomials $U_m$ of the second kind for $m=0,1,2,\dots$.

The polynomials generated from $E_m$ functions of $A_1$, form a standard monomial sequence $y^m$, $m=0,1,2\dots$, which is the basis for the vector space of polynomials.

$C$- and $S$-orbit functions are orthogonal on the interval $F=[0,1]$ (see (\[cont\_orthog c funcs\]) and (\[cont\_orthog s funcs\])) what implies the orthogonality of the corresponding polynomials.

Comparing the properties of one-dimensional orbit functions with properties of Chebyshev polynomials, we conclude that there is a one-to-one correspondence between the Chebyshev polynomials and the orbit functions.

### Orbit 