Sums of products involving power sums of $\phi (n)$ integers

Abstract

A sequence of rational numbers as a generalization of the sequence of Bernoulli numbers is introduced. Sums of products involving the terms of this generalized sequence are then obtained using an application of the Faà di Bruno's formula. These sums of products are analogous to the higher order Bernoulli numbers and are used to develop the closed form expressions for the sums of products involving the power sums ${\Psi }_k(x,n):=\sum _{d|n}\mu (d)d^k{S}_k(\frac{x}{d}),n\in {\mathbb{Z}}^{+}$ which are defined via the Möbius function $\mu$ and the usual power sum $S_k(x)$ of a real or complex variable $x.$ The power sum $S_k(x)$ is expressible in terms of the well known Bernoulli polynomials by $S_k(x):=\frac{B_{k+1}(x+1)-B_{k+1}(0)}{k+1}.$