Sums of products of power sums

Abstract

For any two arithmetic functions $f,g$ let $\bullet$ be the commutative and associative arithmetic convolution $(f\bullet g)(k):={\sum }_{m=0}^k\left(\begin{array}{c}km\end{array}\right)f(m)g(k-m)$ 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 ${\mathcal{S}}_x(k):=\frac{{\mathcal{B}}_{x+1}(k+1)-{\mathcal{B}}_1(k+1)}{k+1},x\ne 0$ 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 ${\mathcal{S}}_x^N(k)(x)$ 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.

arXiv commentThis paper has been withdrawn by the author because of lot many errors

References in corpus (1)

• Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers