# Definition:Bernoulli Numbers/Archaic Form

## Definition

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

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

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

### Definition 2

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

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

### Sequence of Bernoulli Numbers: Archaic Form

The sequence of old style Bernoulli numbers begins:

 $\ds B_1^*$ $=$ $\ds \dfrac 1 6$ $\ds = B_2$ $\ds B_2^*$ $=$ $\ds \dfrac 1 {30}$ $\ds = -B_4$ $\ds B_3^*$ $=$ $\ds \dfrac 1 {42}$ $\ds = B_6$ $\ds B_4^*$ $=$ $\ds \dfrac 1 {30}$ $\ds = -B_8$ $\ds B_5^*$ $=$ $\ds \dfrac 5 {66}$ $\ds = B_{10}$ $\ds B_6^*$ $=$ $\ds \dfrac {691} {2730}$ $\ds = -B_{12}$ $\ds B_7^*$ $=$ $\ds \dfrac 7 6$ $\ds = B_{14}$ $\ds B_8^*$ $=$ $\ds \dfrac {3617} {510}$ $\ds = -B_{16}$ $\ds B_9^*$ $=$ $\ds \dfrac {43 \, 867} {798}$ $\ds = B_{18}$ $\ds B_{10}^*$ $=$ $\ds \dfrac {174 \, 611} {330}$ $\ds = -B_{20}$ $\ds B_{11}^*$ $=$ $\ds \dfrac {854 \, 513} {138}$ $\ds = B_{22}$ $\ds B_{12}^*$ $=$ $\ds \dfrac {236 \, 364 \, 091} {2730}$ $\ds = -B_{24}$

where $B_2, B_4, \ldots$ are the standard form Bernoulli numbers.