Malaysian Journal of Mathematical Sciences, September 2020, Vol. 14, No. 3
On Cyclotomic Polynomial Coefficients
Andrica, D. and Bagdasar, O.
Corresponding Email: dandrica@math.ubbcluj.ro
Received date: 30 October 2019
Accepted date: 30 August 2020
Abstract:
For a positive integer \(n\geq 1\) the \(n\)-th cyclotomic polynomial is defined by \(\Phi_{n}(z)=\prod_{\zeta^{n}=1}(z-\zeta)\), where \(\zeta\) are the primitive \(n\)-th roots of unity. These polynomials are known to possess many interesting properties. In this article we establish an integral formula for the coefficients of the cyclotomic polynomial, we then discuss the direct and alternate sums of coefficients, as well as the mid-term of \(\Phi_{n}(z)\). Finally, these results are used in the computation of certain trigonometric integrals.
Keywords: Euler totient function, cyclotomic polynomial, coefficients of cyclotomic polynomials, integral formula, direct sum of coefficients, mid-term coefficient, alternate sum of coefficients
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