Suppose $m$ is a positive integer, and we know that the following identity holds for all $0<x<2\pi$: $$\left(\sum_{n=1}^{\infty}\frac{\sin(n\,x)}{n^2}\right)^m=c^{\small(m)}_0+\sum_{n=1}^{\infty}c^{\small(m)}_n\cos(n\,x)+\sum_{n=1}^{\infty}s^{\small(m)}_n\sin(n\,x),$$ where $c^{\small(m)}_n,\,s^{\small(m)}_n$ are some coefficients not depending on $x$ (but depending on $m$ and $n$).
Is it possible to find a general formula for $c^{\small(m)}_n,\,s^{\small(m)}_n$?
If we cannot find a general formula, is it possible to find particular formulas for small fixed values of $m=2,3,...$?
