Sums of products of power sums

Abstract

For any two arithmetic functions $f,g$ let $\bullet$ be the commutative and associative arithmetic convolution $(f\bullet g)(k):=\sum_{m=0}^k \left( \begin{array}{c} k m \end{array} \right)f(m)g(k-m)$ and for any $n\in\mathbb{N},$ $f^n=f\bullet \cdots\bullet f$ be $n-$fold product of $f\in \mathcal{S}.$ For any $x\in\mathbb{C},$ let $\mathcal{S}_0=e$ be the multiplicative identity of the ring $(\mathcal{S},\bullet,+)$ and $\mathcal{S}_x(k):=\frac{\mathcal{B}_{x+1}(k+1)-\mathcal{B}_{1}(k+1)}{k+1},~x\neq 0$ denote the power sum defined by Bernoulli polynomials $\mathcal{B}_x(k)=B_k(x).$ We consider the sums of products $\mathcal{S}_x^N(k),~N\in\mathbb{N}_0.$ A closed form expression for $\mathcal{S}^N_x(k)(x)$ generalizing the classical Faulhaber formula, is derived. Furthermore, some properties of $\alpha-$Euler numbers \cite{JS9}(a variant of Apostol Bernoulli numbers) and their sums of products, are considered using which a closed form expression for the sums of products of infinite series of the form $\eta_\alpha(k):=\sum_{n=0}^{\infty}\alpha^n n^k,~0<|\alpha|<1,~k\in\mathbb{N}_0$ and the related Abel sums, is obtained which in particular, gives a closed form expression for well known Bernoulli numbers. A generalization of the sums of products of power sums to the sums of products of alternating power sums is also obtained. These considerations generalize in a unified way to define sums of products of power sums for all $k\in\mathbb{N}$ hence connecting them with zeta functions.