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, 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 $\left\lvert{x}\right\rvert < \pi$


Sequence of Bernoulli Numbers: Archaic Form

The sequence of old style Bernoulli numbers begins:

\(\displaystyle B_1^*\) \(=\) \(\displaystyle \dfrac 1 6\)
\(\displaystyle B_2^*\) \(=\) \(\displaystyle \dfrac 1 {30}\)
\(\displaystyle B_3^*\) \(=\) \(\displaystyle \dfrac 1 {42}\)
\(\displaystyle B_4^*\) \(=\) \(\displaystyle \dfrac 1 {30}\)
\(\displaystyle B_5^*\) \(=\) \(\displaystyle \dfrac 5 {66}\)
\(\displaystyle B_6^*\) \(=\) \(\displaystyle \dfrac {691} {2730}\)
\(\displaystyle B_7^*\) \(=\) \(\displaystyle \dfrac 7 6\)
\(\displaystyle B_8^*\) \(=\) \(\displaystyle \dfrac {3617} {510}\)
\(\displaystyle B_9^*\) \(=\) \(\displaystyle \dfrac {43 \, 867} {798}\)
\(\displaystyle B_{10}^*\) \(=\) \(\displaystyle \dfrac {174 \, 611} {330}\)
\(\displaystyle B_{11}^*\) \(=\) \(\displaystyle \dfrac {854 \, 513} {138}\)
\(\displaystyle B_{12}^*\) \(=\) \(\displaystyle \dfrac {236 \, 364 \, 091} {2730}\)


Also see


Sources