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

2025/01/27 by Mamoru Doi, Doi, Mamoru
Computer Science · Mathematics · Physics and Astronomy · #11R04 #12D10 #33C45 #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Theories and Applications #Coding theory and cryptography #FOS: Mathematics #General Mathematics (math.GM)

paper · pdf · doi:10.48550/arxiv.2501.16478

openalex publication_date 2025/01/27 · openalex created_date 2025/01/30 · openalex updated_date 2026/07/29

Abstract

In this paper we consider the minimal polynomial ψ_n(x) of 2cos (2π/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)± 2=2(T_n(x/2)± 1) are divisible by the square of either of these polynomials. Then by appropriately removing unnecessary factors from these polynomials, we can easily calculate ψ_n(x) without recursion, which improves Barnes' result in 1977. As an appendix, we give a compact table of the minimal polynomials ψ_n(x) of 2cos (2π/n) for n\leqslant 120.

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