FACTORING VARIANTS OF CHEBYSHEV POLYNOMIALS WITH MINIMAL POLYNOMIALS OF $\mathbf {cos}\boldsymbol {({2\pi }/{d})}

• Published in: Bulletin of the Australian Mathematical Society 2022-12, Vol.106 (3), p.448-457
• Main Author: WOLFRAM, D. A.
• Format: Article
• Language: English
• Subjects: Chebyshev approximation; Partial differential equations; Polynomials; Temperature
• ISSN: 0004-9727; 1755-1633
• DOI: 10.1017/S0004972722000235

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, in terms of the minimal polynomials of $\cos ({2\pi }{/d})$ . The method of proof is based on earlier work, D. A. Wolfram, [‘Factoring variants of Chebyshev polynomials of the first and second kinds with minimal polynomials of $\cos ({2 \pi }/{d})$ ’, Amer. Math. Monthly 129 (2022), 172–176] for factoring variants of Chebyshev polynomials of the first and second kinds. We extend this to show that, in general, similar variants of Chebyshev polynomials of the fifth and sixth kinds, $X_n(x) \pm 1$ and $Y_n(x) \pm 1$ , do not have factors that are minimal polynomials of $\cos ({2\pi }/{d})$ .