On polynomials with simple trigonometric formulas (Q1777862)

The author proves that \(\Im ((\cot \alpha +i)^n)=\frac{\sin(n.\alpha)}{\sin^n\alpha}\) and \(\Re ((\cot \alpha +i)^n)=\frac{\cos(n.\alpha)}{\sin^n\alpha}\). Similar identities are obtained for Chebyshev polynomials. He also proves that the both sequences \(\{\Im ((x +i)^n)\}_{n=1}^\infty\) and \(\{\Re ((x +i)^n)\}_{n=1}^\infty\) do not create a system of orthogonal polynomials. Some relations among \(\Im ((x +i)^n)\), \(\Re ((x +i)^n)\) and Chebyshev polynomials are included.