Sums of products of power sums

TL;DR

This paper derives closed-form formulas for sums of products of power sums related to Bernoulli polynomials, Euler numbers, and zeta functions, extending classical formulas and exploring their properties and generalizations.

Contribution

It introduces a unified approach to sums of power sums, generalizing Faulhaber's formula and connecting with Bernoulli, Euler, and zeta functions, including alternating sums and infinite series.

Findings

Derived closed-form expressions for sums of power sums.

Connected sums of power sums with Bernoulli and Euler numbers.

Extended formulas to alternating power sums and related zeta functions.

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.