Definition:Bernoulli Numbers

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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.


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.


Sources

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