Shifted sums of the Bernoulli numbers, reciprocity, and denominators

TL;DR

This paper investigates the properties of shifted sums of Bernoulli numbers, proving their non-vanishing in general and deriving explicit formulas for their denominators, thus extending classical reciprocity relations.

Contribution

It establishes the non-vanishing of shifted Bernoulli sums and provides a new explicit formula for their denominators, linking to classical reciprocity and von Staudt--Clausen relations.

Findings

Proves non-vanishing of the shifted Bernoulli sums except for specific cases.

Derives an explicit product formula for the denominators of these sums.

Connects the properties of these sums to classical reciprocity and Bernoulli number theory.

Abstract

We consider the numbers $\mathcal{B}_{r,s} = (\mathbf{B}+1)^r \mathbf{B}^s$ (in umbral notation $\mathbf{B}^n = \mathbf{B}_n$ with the Bernoulli numbers) that have a well-known reciprocity relation, which is frequently found in the literature and goes back to the 19th century. In a recent paper, self-reciprocal Bernoulli polynomials, whose coefficients are related to these numbers, appeared in the context of power sums and the so-called Faulhaber polynomials. The numbers $\mathcal{B}_{r,s}$ can be recursively expressed by iterated sums and differences, so it is not obvious that these numbers do not vanish in general. As a main result among other properties, we show the non-vanishing of these numbers, apart from exceptional cases. We further derive an explicit product formula for their denominators, which follows from a von Staudt--Clausen type relation.