Remarks on the coefficients of ternary cyclotomic and inverse cyclotomic polynomials

Abstract

Using a formula attributed to Von Sterneck, we present a method for deriving recursive relations for the coefficients of ternary cyclotomic polynomials Φ_n(x) and ternary inverse cyclotomic polynomials Ψ_n(x). We consider n = pqr with p < q < r are primes. As an application, we determine exact values for the first (p + 1) coefficients. For an infinite subfamily, the first r coefficients are computed. In the case of ternary cyclotomic polynomials, we identify an explicit infinite family whose r-th coefficient is equal to −2. Additionally, we include a section on numerical simulations that demonstrates the computation of non-flat coefficients in specific cases for Φ₁₀₅ and Ψ₅₆₁.