Polynomial sequences related to Chebyshev polynomials and the minimal polynomial of $2\cos (2\pi /n)$

Abstract

In this paper we consider the minimal polynomial $\psi_n(x)$ of $2\cos (2\pi /n)$. We introduce some polynomial sequences with the same recurrence relation as the rescaled Chebyshev polynomials $t_n(x)=2\, T_n(x/2)$ of the first kind, which turn out to be related to those of various kinds, all coming from those of the second kind. We see that $t_n(x)\pm 2=2(T_n(x/2)\pm 1)$ are divisible by the square of either of these polynomials. Then by appropriately removing unnecessary factors from these polynomials, we can easily calculate $\psi_n(x)$ without recursion, which improves Barnes' result in 1977. As an appendix, we give a compact table of the minimal polynomials $\psi_n(x)$ of $2\cos (2\pi /n)$ for $n\leqslant 120$.