2022/01/06 by D. A. Wolfram · 1 citation
Mathematics · Computer Science · #Advanced Differential Equations and Dynamical Systems #Coding theory and cryptography #Mathematical functions and polynomials #Chebyshev polynomials #Mathematics #Factorization #Integer (computer science) #Polynomial #Combinatorics #Factorization of polynomials #Discrete mathematics #Classical orthogonal polynomials #Orthogonal polynomials #Pure mathematics #Algorithm #Mathematical analysis #Computer science #Matrix polynomial
paper · doi:10.1080/00029890.2022.2005391
openalex publication_date 2022/01/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
We solve the open problem of Y. Z. Gürtaş of factoring polynomials Un(x)±1, where Un(x) are Chebyshev polynomials of the second kind. While the factors of Un(x)2−1 were known, the pattern of their mapping to Un(x)+1 and Un(x)−1 was not, and it was intriguing. Each factor is a polynomial of the form Ψd(x) with integer coefficients whose roots are cos (2πk/d). We then consider the analogous problem for Chebyshev polynomials of the first kind by deriving a factorization of Tn(x)2−1. This factorization enables the method used to solve the open problem to be applied to this one, resulting in a more direct proof.