# Definition:Bernoulli Numbers

## Definition

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

### Generating Function

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

### Recurrence Relation

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

### Sequence

The sequence of Bernoulli numbers begins:

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

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

 $\displaystyle \frac x {e^x - 1}$ $=$ $\displaystyle 1 - \frac x 2 + \sum_{n \mathop = 1}^\infty \left({-1}\right)^{n - 1} \frac{B_n^* x^{2 n} } {\left({2 n}\right)!}$ $\displaystyle$ $=$ $\displaystyle 1 - \frac x 2 + \frac{B_1^* x^2} {2!} - \frac{B_2^* x^4} {4!} + \frac{B_3^* x^6} {6!} - \cdots$

for $x \in \R$ such that $\left\lvert{x}\right\rvert < 2 \pi$

### Definition 2

$\displaystyle 1 - \frac x 2 \cot \frac x 2 = \sum_{n \mathop = 1}^\infty \frac{B_n^* x^{2 n} } {\left({2 n}\right)!}$
 $\displaystyle 1 - \frac x 2 \cot \frac x 2$ $=$ $\displaystyle \sum_{n \mathop = 1}^\infty \frac{B_n^* x^{2 n} } {\left({2 n}\right)!}$ $\displaystyle$ $=$ $\displaystyle \frac{B_1^* x^2} {2!} + \frac{B_2^* x^4} {4!} + \frac{B_3^* x^6} {6!} + \cdots$

for $x \in \R$ such that $\left\lvert{x}\right\rvert < \pi$

## 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.