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''$${\mathrm{\Gamma }}_{n,m}(\xi )=\cot \text{(}{\displaystyle \frac{(\xi +m)\pi }{n}}\text{)}\text{and}{\mathrm{\Delta }}_{n,m}(\xi )=\tan \text{(}{\displaystyle \frac{(\xi +m)\pi }{n}}\text{)}(0<\xi <1)$$($n=1,2,\dots$; $m=0,1,\dots ,n-1$ and $m\ne (n-1)/2$ when $\xi =1/2$ and $n$ is odd in the case of ${\mathrm{\Delta }}_{n,m}$) are obtained from the linear combination of the Chebyshev polynomials of the first and second kind.