Curious congruences for cyclotomic polynomials.

Let Φ n (k) (x) be the kth derivative of the nth cyclotomic polynomial. We are interested in the values Φ n (k) (1) for fixed positive integers n. D. H. Lehmer proved that Φ n (k) (1) / Φ n (1) is a polynomial of the Euler totient function ϕ (n) and the Jordan totient functions and gave its explicit formula. In this paper, we give a quick proof that Φ n (k) (1) / Φ n (1) is a polynomial of them without giving the explicit form. In the final section, we deduce some curious congruences: 2 Φ n (3) (1) is divisible by ϕ (n) - 2 . Moreover, if k is greater than 1, then Φ n (2 k + 1) (1) is divisible by ϕ (n) - 2 k . 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 Φ n (k) (1) as a Z -linear combination of the coefficients in the minimal polynomial of 2 cos (2 π / n) - 2 . As a supplement, we show the monotonic increasing property of Φ n (x) on [ 1 , ∞) in two ways. [ABSTRACT FROM AUTHOR]