Factoring Variants of Chebyshev Polynomials with Minimal Polynomials of $\cos(\frac{2\pi}{d})$

TL;DR

This paper presents methods for factoring specific variants of Chebyshev polynomials of the third and fourth kinds using minimal polynomials of cosines, extending previous work on first and second kinds, and shows non-factorizability for fifth and sixth kinds.

Contribution

It introduces new factorization results for Chebyshev polynomial variants of the third and fourth kinds and demonstrates the limitations for fifth and sixth kinds.

Findings

Factoring formulas for $V_n(x) \u002b 1$, $V_n(x) \u002d 1$, $W_n(x) \u002b 1$, and $W_n(x)  1$.

No factorizations exist for variants of fifth and sixth kinds using minimal polynomials of os(\u03c0/d).

Extension of Wolfram's methods to Chebyshev polynomials of the third and fourth kinds.

Abstract

We solve the problem of factoring polynomials $V_n(x) \pm 1$ and $W_n(x) \pm 1$ where $V_n(x)$ and $W_n(x)$ are Chebyshev polynomials of the third and fourth kinds. The method of proof is based on previous work by Wolfram [12] for factoring variants of Chebyshev polynomials of the first and second kinds, $T_n(x) \pm 1$ and $U_n(x) \pm 1$. We also show that, in general, there are no factorizations of variants of Chebyshev polynomials of the fifth and sixth kinds, $X_n(x) \pm 1$ and $Y_n(x) \pm 1$ using minimal polynomials of $\cos(\frac{2\pi}{d})$.