Chebyshev polynomials

Remarks

The first values are $T_2 = 2x^2 - 1$, $T_3 = 4x^3 - 3x$ and $T_4 = 8x^4 - 8x^2 + 1$. Check for $T_2$: $\cos(2\theta) = 2 \cos^2 \theta - 1$. For $n \ge 1$, $T_n$ has degree $n$ and leading coefficient $2^{n-1}$. Its roots are the $n$ numbers $\cos \frac{(2k - 1)\pi}{2n}$, $k = 1, \dots, n$, all in $(-1, 1)$. On $[-1, 1]$ one has $\abs{T_n(x)} \le 1$, and the $T_n$ are orthogonal for the weight $(1 - x^2)^{-1/2}$: $\int_{-1}^{1} T_m(x) T_n(x) (1 - x^2)^{-1/2} \, dx = 0$ for $m \ne n$. They also satisfy $T_m(T_n(x)) = T_{mn}(x)$. Their roots are good nodes for polynomial interpolation on $[-1, 1]$.