A shortened recurrence relation for the Bernoulli numbers

TL;DR

This paper introduces a new, efficient recurrence relation for Bernoulli numbers that simplifies their computation by deriving higher even-indexed Bernoulli numbers from lower ones, improving computational methods.

Contribution

The paper presents a novel recurrence relation for Bernoulli numbers derived from Kuo's result, enabling more efficient calculations of even-indexed Bernoulli numbers.

Findings

Allows computation of B_{4n} and B_{4n+2} from B_0, B_2, ..., B_{2n}

Simplifies calculation process compared to previous formulas

Provides a potentially more efficient method for Bernoulli number computation

Abstract

In this note, starting with a little-known result of Kuo, I derive a recurrence relation for the Bernoulli numbers $B_{2 n}$, $n$ being any positive integer. This new recurrence seems advantageous in comparison to other known formulae since it allows the computation of both $B_{4 n}$ and $B_{4 n +2}$ from only $B_0, B_2,..., B_{2n}$.