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Abstract

The $n^{th}$ cyclotomic polynomial $\Phi_n(x)$ is the minimal polynomial of
an $n^{th}$ primitive root of unity. Its coefficients are the subject of
intensive study and some formulas are known for them. Here we are interested in
formulas which are valid for all natural numbers $n$. In these a host of famous
number theoretical objects such as Bernoulli numbers, Stirling numbers of both
kinds and Ramanujan sums make their appearance, sometimes even at the same
time!
In this paper we present a survey of these formulas which until now were
scattered in the literature and introduce an unified approach to derive some of
them, leading also to shorter proofs as a by-product. In particular, we show
that some of the formulas have a more elegant reinterpretation in terms of Bell
polynomials. This approach amounts to computing the logarithmic derivatives of
$\Phi_n$ at certain points. Furthermore, we show that the logarithmic
derivatives at $\pm 1$ of any Kronecker polynomial (a monic product of
cyclotomic polynomials and a monomial) satisfy a family of linear equations
whose coefficients are Stirling numbers of the second kind. We apply these
equations to show that certain polynomials are not Kronecker. In particular, we
infer that for every $k\ge 4$ there exists a symmetric numerical semigroup with
embedding dimension $k$ and Frobenius number $2k+1$ that is not cyclotomic,
thus establishing a conjecture of Alexandru Ciolan, Pedro Garc\'ia-S\'anchez
and the second author. In an appendix Pedro Garc\'ia-S\'anchez shows that for
every $k\ge 4$ there exists a symmetric non-cyclotomic numerical semigroup
having Frobenius number $2k+1.$