Factoring Variants of Chebyshev Polynomials of the First and Second Kinds with Minimal Polynomials of cos(2π/d)

Abstract

We solve the open problem of Y. Z. Gürtaş of factoring polynomials $U_n(x)\pm 1$, where $U_n(x)$ are Chebyshev polynomials of the second kind. While the factors of $U_n{(x)}^2-1$ were known, the pattern of their mapping to $U_n(x)+1$ and $U_n(x)-1$ was not, and it was intriguing. Each factor is a polynomial of the form ${\Psi }_d(x)$ with integer coefficients whose roots are $\text{cos}\hspace{0.17em}(2\pi k/d)$. We then consider the analogous problem for Chebyshev polynomials of the first kind by deriving a factorization of $T_n{(x)}^2-1$. This factorization enables the method used to solve the open problem to be applied to this one, resulting in a more direct proof.

MSC::

• Primary 12E10
• Secondary 12D05

ACKNOWLEDGMENTS

I thank the referees and Susan Jane Colley for their helpful suggestions. I am grateful to the College of Engineering & Computer Science at The Australian National University for research support.