Definition:Bernoulli Numbers/Archaic Form

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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.


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 $\size x < \pi$


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.


Also see


Sources

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