A New Expression for the Bernoulli Numbers and its Applications
Levent Kargın, Merve Mutluer
TL;DR
This work presents a new expression for a finite convolution of Stirling numbers with harmonic numbers that collapses to a Bernoulli-number formula, enabling a direct derivation of Bernoulli recurrences and cumulative sums. By proving the main identity $\sum_{k=j}^{n} (-1)^{k-j} {n \brace k} [k j] H_k = ( \binom{n}{j} - 1) \frac{B_{n-j}}{n-j}$, the authors reprove Agoh's linear recurrence and obtain a new Bernoulli recurrence, while also relating cumulative sums to di-Bernoulli numbers via $\sum_{j=0}^{n} B_j = \mathbb{B}_{n}^{(2)}(1) + B_n - \mathbb{B}_{n}^{(2)} - 1$. They further derive congruences for sums of Bernoulli and Euler numbers modulo odd primes and squares, highlighting arithmetic structure and potential number-theoretic applications.
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.