Definition:Bernoulli Numbers

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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.


Sources