A generalisation of the Bernoulli numbers from the discrete to the continuous

TL;DR

This paper extends Bernoulli numbers to a continuous index, broadening their applicability from discrete to continuous mathematical contexts.

Contribution

It introduces a novel generalization of Bernoulli numbers allowing for continuous indices, which was not previously explored.

Findings

Defined a continuous analogue of Bernoulli numbers.

Established properties and potential applications of the generalized numbers.

Bridged discrete Bernoulli numbers with continuous mathematical analysis.

Abstract

We generalise the Bernoulli numbers to include the case where the index may be a continuous variable.