# Definition:Bernoulli Numbers

## Definition

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

### Generating Function

$\ds \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$

### Recurrence Relation

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

or equivalently:

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

### Sequence

The sequence of Bernoulli numbers begins:

 $\ds B_0$ $=$ $\ds 1$ $\ds B_1$ $=$ $\, \ds - \,$ $\ds \dfrac 1 2$ $\ds B_2$ $=$ $\ds \dfrac 1 6$ $\ds B_4$ $=$ $\, \ds - \,$ $\ds \dfrac 1 {30}$ $\ds B_6$ $=$ $\ds \dfrac 1 {42}$ $\ds B_8$ $=$ $\, \ds - \,$ $\ds \dfrac 1 {30}$ $\ds B_{10}$ $=$ $\ds \dfrac 5 {66}$ $\ds B_{12}$ $=$ $\, \ds - \,$ $\ds \dfrac {691} {2730}$

The odd index Bernoulli numbers, apart from $B_1$, are all equal to $0$.

## Archaic Form

A different definition of the Bernoulli numbers can be found in older literature.

Usually denoted with the symbol ${B_n}^*$, they are considered archaic, and will not be used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

### Definition 1

 $\ds \frac x {e^x - 1}$ $=$ $\ds 1 - \frac x 2 + \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {B_n^* x^{2 n} } {\paren {2 n}!}$ $\ds$ $=$ $\ds 1 - \frac x 2 + \frac {B_1^* x^2} {2!} - \frac {B_2^* x^4} {4!} + \frac {B_3^* x^6} {6!} - \cdots$

for $x \in \R$ such that $\size x < 2 \pi$

### Definition 2

 $\ds 1 - \frac x 2 \cot \frac x 2$ $=$ $\ds \sum_{n \mathop = 1}^\infty \frac {B_n^* x^{2 n} } {\paren {2 n}!}$ $\ds$ $=$ $\ds \frac {B_1^* x^2} {2!} + \frac {B_2^* x^4} {4!} + \frac {B_3^* x^6} {6!} + \cdots$

for $x \in \R$ such that $\size x < \pi$

## Also see

• Results about the Bernoulli Numbers can be found here.

## Source of Name

This entry was named for Jacob Bernoulli.

## Historical Note

The Bernoulli numbers were introduced by Jacob Bernoulli in his investigations into the power series expansion of the tangent function.