An Identity involving the Cyclotomic Polynomials

2020/10/06 by Pablo L. De Nápoli, De Nápoli, Pablo L.
Mathematics · #11A25 #11C08 #Analytic Number Theory Research #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2010.02668

openalex publication_date 2020/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present an elementary identity for the cyclotomic polynomials Φ_n(X) which reflects a kind of multiplicative property of Φ_n(X) as a function of n, and we explore its connections with the properties of other arithmetical functions. Important Note: In the first version of this article uploaded to the arXiv, it is said that this result seemed to be new. However, after that, I have learned that this identity (in Theorem 1.1) has previously appeared in Cheng, C. C. A., McKay, J. H., & Wang, S. S. S. (1995). Resultants of cyclotomic polynomials. Proceedings of the American Mathematical Society, 123(4), 1053-1059. (with a different proof).

Related

• Generalized cyclotomic polynomials associated with regular systems of divisors and arbitrary sets of positive integers
• Curious congruences for cyclotomic polynomials II
• A Survey on Coefficients of Cyclotomic Polynomials
• Explicit expression for a family of ternary cyclotomic polynomials
• Coefficients of cyclotomic polynomials