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

Part of: Real and complex fields General field theory

Published online by Cambridge University Press:  21 March 2022

D. A. WOLFRAM Open the ORCID record for D. A. WOLFRAM [Opens in a new window]
D. A. WOLFRAM*
Affiliation:
College of Engineering and Computer Science, The Australian National University, Canberra, ACT 0200, Australia
*
e-mail: [email protected]
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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, 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 $\cos ({2\pi }/{d})$ .

Keywords

Chebyshev polynomialsfactorisationminimal polynomial

MSC classification

Primary: 12E10: Special polynomials
Secondary: 12D05: Polynomials
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 106 , Issue 3 , December 2022 , pp. 448 - 457
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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