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

TL;DR

This paper introduces a generalized sequence related to Bernoulli numbers, derives sums of products involving this sequence and power sums, and provides closed-form formulas for sums involving the Möbius function and power sums.

Contribution

It develops a new rational sequence generalizing Bernoulli numbers and derives explicit formulas for sums involving power sums and the Möbius function using Faà di Bruno's formula.

Findings

Introduces a generalized sequence extending Bernoulli numbers.

Derives closed-form expressions for sums involving power sums and the Möbius function.

Provides formulas analogous to higher order Bernoulli numbers.

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 Fa\`a 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 $\displaystyle \Psi_k(x,n):=\sum_{d|n}\mu(d)d^k S_k(\frac{x}{d}), n\in\mathbb{Z}^+$ which are defined via the M\"obius 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 $\displaystyle S_k(x):=\frac{B_{k+1}(x+1)-B_{k+1}(0)}{k+1}.$