An application for the Chebyshev polynomials

Abstract

Two sequences of polynomials for which all zeros, regardless of degree $n$, can be given by the following "simple formulae''$$\Gamma_{n,m}(\xi)=\cot\(\dfrac{(\xi+m)\pi}n\)\quad\text{and}\quad\Delta_{n,m}(\xi)=\tan\(\dfrac{(\xi+m)\pi}n\)\quad(0<\xi<1)$$($n=1,2,\dots$; $m=0,1,\dots,n-1$ and $m\ne(n-1)/2$ when $\xi=1/2$ and $n$ is odd in the case of $\Delta_{n,m}$) are obtained from the linear combination of the Chebyshev polynomials of the first and second kind.