Curious congruences for cyclotomic polynomials

Let Phi((k))(n) (x) be the kth derivative of the nth cyclotomic polynomial. We are interested in the values Phi((k))(n) (1) for fixed positive integers n. D. H. Lehmer proved that Phi((k))(n) (1) /Phi(n) (1) is a polynomial of the Euler totient function phi(n) and the Jordan totient functions and gave its explicit formula. In this paper, we give a quick proof that Phi((k))(n) (1) / Phi(n) (1) is a polynomial of them without giving the explicit form. In the final section, we deduce some curious congruences: 2 Phi((3))(n) (1) is divisible by phi(n) - 2. Moreover, if k is greater than 1, then Phi((2k+1))(n) (1) is divisible by phi(n) - 2k. The proof depends on a new combinatorial identity for general self-reciprocal polynomials over Z, which gives rise to a formula that expresses the value Phi((k))(n) (1) as a Z-linear combination of the coefficients in the minimal polynomial of 2 cos(2 pi/n) - 2. As a supplement, we show the monotonic increasing property of Phi(n) (x) on [1, infinity) in two ways.