An introduction to the Bernoulli function

2020/09/11 by Peter H. N. Luschny, Luschny, Peter H. N.
Mathematics · #11B68 (Primary) #11M35 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #History and Overview (math.HO) #History and Theory of Mathematics

paper · pdf · doi:10.48550/arxiv.2009.06743

openalex publication_date 2020/09/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We explore a variant of the zeta function interpolating the Bernoulli numbers based on an integral representation suggested by J. Jensen. The Bernoulli function B(s, v) = - s ζ(1-s, v) can be introduced independently of the zeta function if it is based on a formula first given by Jensen in 1895. We examine the functional equation of B(s, v) and its representation by the Riemann ζ and ξ function, and recast classical results of Hadamard, Worpitzky, and Hasse in terms of B(s, v). The extended Bernoulli function defines the Bernoulli numbers for odd indices basing them on rational numbers studied by Euler in 1735 that underlie the Euler and André numbers. The Euler function is introduced as the difference between values of the Hurwitz-Bernoulli function. The André function and the Seki function are the unsigned versions of the extended Euler resp. Bernoulli function.

Citations

• Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. Volume I
• On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers

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