Factoring Variants of Chebyshev Polynomials with Minimal Polynomials of cos((2π)/(d))

2021/06/29 by D. A. Wolfram · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Chebyshev polynomials #Classical orthogonal polynomials #Combinatorics #Discrete mathematics #Factoring #Mathematical analysis #Mathematical functions and polynomials #Mathematics #Orthogonal polynomials #Pure mathematics #math.CA #math.RA #msc:12D05 #msc:12E10

paper · pdf · open access · doi:10.1017/s0004972722000235

published in Bulletin of the Australian Mathematical Society 106(3), 448-457 (Cambridge University Press) · 9 pages

arxiv created 2021/06/29 · openalex created_date 2021/07/05 · openalex publication_date 2022/03/21 · arxiv updated 2022/03/22 · openalex updated_date 2026/08/05

Abstract

We solve the problem of factoring polynomials V_n(x) ± 1 and W_n(x) ± 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) ± 1 and U_n(x) ± 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) ± 1 and Y_n(x) ± 1 using minimal polynomials of cos((2π)/(d)).

Citations

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