For a natural number n, let $r(x)$ be the polynomial $$r(x)=\prod_{k=1}^n(x-2\sin(\frac{\pi k}{2n+1})).$$ Then $-xr(2x)r(-2x)$ is Chebyshev polynomial of the first kind with integer coefficients. There are two questions with applications to solving difference equations:
Is there a closed formula for the coefficients of $r(x)$?
Let $$q(x^2)=r(x)+r(-x)$$ and $$xp(x^2)=r(x)-r(-x),$$ then the roots of $p(x)$ and $q(x)$ are negative and interlace. Is there an explicit formula for the roots?