\frac{\pi}{2},\quad n=m\ne 0,
\end{array}
\right.
\\
\int\limits^1_{-1}\sqrt{1-x^2}U_n(x)U_m(x){\rm d}x=
\left\{
\begin{array}{l}
0,\quad n\ne m,\\
\frac{\pi}{2},\quad n=m.
\end{array}
\right.\end{gathered}$$

There are other useful relations between Chebyshev polynomials of the first and second kind. $$\begin{gathered}
\frac{\rm d}{{\rm d}x} T_n(x)=n U_{n-1}(x),\quad n=1,2,3,\dots \label{differentiation_Tn}
\\
T_n(x)=\frac 12 (U_{n}(x)-U_{n-2}(x)),\quad n=2,3,\dots \label{difference_Un_Un-2}
\\
T_{n+1}(x)=xT_n(x)-(1-x^2)U_{n-1},\quad n=1,2,3,\dots \label{difference_xTn_Un-1}
\\
T_{n}(x)=U_n(x)-xU_{n-1},\quad n=1,2,3,\dots \label{difference_Un_xUn-1}\end{gathered}$$

### Trigonometric form of Chebyshev polynomials

Using trigonometric variable $x=\cos y$, polynomials of the first kind become $$\begin{gathered}
\label{trigon_T}
T_n(x)=T_n(\cos y)=\cos(ny), \quad n=0,1,2,\dots\end{gathered}$$ and polynomials of the second kind are written as $$\begin{gathered}
\label{trigon_U}
U_n(x)=U_n(\cos y)=\frac{\sin((n+1)y)}{\sin y}, \quad n=0,1,2,\dots\end{gathered}$$

For example, the first few lowest polynomials are $$\begin{gathered}
T_0(x)=T_0(\cos y)=\cos(0y)=1, \quad
T_1(x)=T_1(\cos y)=\cos(y)=x, \\
T_2(x)=T_2(\cos y)=\cos(2y)=\cos^2y-\sin^2y=2cos^2y-1=2x^2-1; \quad \\
U_0(x)=U_0(\cos y)=\frac{\sin y}{\sin y}=1, \quad
U_1(x)=U_1(\cos y)=\frac{\sin(2y)}{\sin y}=2\cos y=2x, \\
U_2(x)=U_2(\cos y)=\frac{\sin(3y)}{\sin y}=\frac{\sin(2y)\cos y+\sin y \cos(2y)}{\sin y}
=4\cos^2y-1=4x^2-1.\end{gathered}$$

Orbit functions of $A_1$ and Chebyshev polynomials
--------------------------------------------------

Let us consider the orbit functions of one variable. There is only one simple Lie algebra of rank 1, namely $A_1$. Our aim is to build the recursion relations in a way that generalizes to higher rank groups, unlike the standard relations of the classical theory presented above.

### Orbit functions of $A_1$ and trigonometric form of $T_n$ and $U_n$

The orbit of 