Chebyshev polynomials of the second kind

Summary

The Chebyshev polynomials of the second kind $U_n$ are the polynomials with $U_n(\cos \theta) \sin \theta = \sin((n + 1)\theta)$. They are the companions of the Chebyshev polynomials of the first kind $T_n$, and satisfy the same three-term recurrence with a different start. They are orthogonal on $[-1, 1]$ for the weight $\sqrt{1 - x^2}$, and they are the Gegenbauer polynomials with parameter $1$.