Definition:Bernoulli Numbers/Recurrence Relation

Definition

The Bernoulli numbers $B_n$ are a sequence of rational numbers defined by the recurrence relation:

$B_n = \begin{cases} 1 & : n = 0 \\ \displaystyle - \sum_{k \mathop = 0}^{n - 1} \binom n k \frac {B_k} {n + 1 - k} & : n > 0 \end{cases}$

or equivalently:

$B_n = \begin{cases} 1 & : n = 0 \\ \displaystyle - \frac 1 {n+1} \sum_{k \mathop = 0}^{n - 1} \binom {n+1} k B_k & : n > 0 \end{cases}$

Illustration

For $n \in \N_{>0}$:

$\displaystyle \sum_{k \mathop = 0}^n \binom {n + 1} k B_k = 0$

Also see

• Equivalence of Definitions of Bernoulli Numbers

• Results about the Bernoulli Numbers can be found here.