Chebyshev polynomials of the second kind

Remarks

Equivalently, $U_n$ is the unique polynomial with $U_n(\cos \theta) \sin \theta = \sin((n + 1)\theta)$ for all real $\theta$. Example: $U_2 = 2x \cdot 2x - 1 = 4x^2 - 1$, and indeed $\sin(3\theta) = \sin \theta \, (3 - 4 \sin^2 \theta) = \sin \theta \, (4 \cos^2 \theta - 1)$. Non-example: the Chebyshev polynomials of the first kind $T_n$ satisfy the same recurrence but start with $T_1 = x$, so they fail the clause $U_1 = 2x$. The polynomial $U_n$ has degree $n$, leading coefficient $2^n$, and roots $\cos \frac{k\pi}{n + 1}$ for $k = 1, \dots, n$. They are orthogonal: $\int_{-1}^{1} U_m(x) U_n(x) \sqrt{1 - x^2} \, dx = \frac{\pi}{2} \delta_{mn}$. They are linked to the first kind by $T_n' = n \, U_{n-1}$ for $n \ge 1$.