Category:Definitions/Bernoulli Numbers

This category contains definitions related to Bernoulli Numbers.
Related results can be found in Category:Bernoulli Numbers.

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:

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