A New Expression for the Bernoulli Numbers and its Applications

TL;DR

This paper introduces a novel expression linking Bernoulli numbers with Stirling and harmonic numbers, leading to new recurrence relations, sum formulas, and congruences, enhancing understanding of Bernoulli number properties.

Contribution

It presents a new finite convolution expression involving Bernoulli, Stirling, and harmonic numbers, and derives new recurrence relations and sum formulas for Bernoulli numbers.

Findings

Reproved Agoh's recurrence relation for Bernoulli numbers

Derived a new recurrence relation for Bernoulli numbers

Established congruences for Bernoulli and Euler number sums

Abstract

This paper shows that a finite discrete convolution involving Stirling numbers of both kinds and harmonic numbers can be expressed in terms of the Bernoulli numbers. As applications of this expression, the linear recurrence relation for the Bernoulli numbers given by Agoh is reproved, and a new recurrence relation for the Bernoulli numbers is obtained. Furthermore, it is shown that a cumulative sum of the Bernoulli numbers can be written in terms of the Bernoulli and di-Bernoulli numbers. Finally, congruences for the sums of the Bernoulli and Euler numbers are established.