6.1.1 Computational examples: Chebyshev roots
The roots of \(T_3(x) = 4x^3 - 3x\) are:
\begin{align*} \xi _0 & = \cos (\pi /6) = \frac{\sqrt{3}}{2} \approx 0.866, \\ \xi _1 & = \cos (\pi /2) = 0, \\ \xi _2 & = \cos (5\pi /6) = -\frac{\sqrt{3}}{2} \approx -0.866. \end{align*}
Power sums:
\begin{align*} p_1 & = \xi _0 + \xi _1 + \xi _2 = 0, \\ p_2 & = \xi _0^2 + \xi _1^2 + \xi _2^2 = \frac{3}{4} + 0 + \frac{3}{4} = \frac{3}{2}. \end{align*}
Note that \(p_2 = \frac{3}{2} = \frac{N}{2^j}\binom {j}{j/2} = \frac{3}{4} \cdot 2 = \frac{3}{2}\) when \(j = 2\).
The roots of \(T_4(x) = 8x^4 - 8x^2 + 1\) are:
\begin{align*} \xi _0 & = \cos (\pi /8) \approx 0.924, \\ \xi _1 & = \cos (3\pi /8) \approx 0.383, \\ \xi _2 & = \cos (5\pi /8) \approx -0.383, \\ \xi _3 & = \cos (7\pi /8) \approx -0.924. \end{align*}
Power sums:
\begin{align*} p_1 & = 0 \quad \text{(sum of symmetric points)}, \\ p_2 & = 2\cos ^2(\pi /8) + 2\cos ^2(3\pi /8) = 2 \cdot \frac{2 + \sqrt{2}}{4} + 2 \cdot \frac{2 - \sqrt{2}}{4} = 2, \\ p_3 & = 0 \quad \text{(odd power, symmetric roots)}. \end{align*}
The roots of \(T_5(x) = 16x^5 - 20x^3 + 5x\) are:
\begin{align*} \xi _0 & = \cos (\pi /10) \approx 0.951, \\ \xi _1 & = \cos (3\pi /10) \approx 0.588, \\ \xi _2 & = \cos (\pi /2) = 0, \\ \xi _3 & = \cos (7\pi /10) \approx -0.588, \\ \xi _4 & = \cos (9\pi /10) \approx -0.951. \end{align*}
Power sums: \(p_1 = 0\), \(p_2 = 5/2\), \(p_3 = 0\), \(p_4 = 15/8\).