Shifted sums of the Bernoulli numbers, reciprocity, and denominators. (English) Zbl 1503.11047

Summary: 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.


11B68 Bernoulli and Euler numbers and polynomials
11B65 Binomial coefficients; factorials; \(q\)-identities
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