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.

Keywords: Chebyshev pomynolials, polynomials, zeros of polynomials.

MSC: 33C45, 33C90

Pages:  105$-$110     

Volume  50 ,  Issue  3$-$4 ,  1998