Definition:Bernoulli Numbers/Archaic Form/Definition 1
Jump to navigation
Jump to search
Definition
The old form Bernoulli numbers ${B_n}^*$ are a sequence of rational numbers defined by the exponential generating function:
\(\ds \frac x {e^x - 1}\) | \(=\) | \(\ds 1 - \frac x 2 + \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {B_n^* x^{2 n} } {\paren {2 n}!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - \frac x 2 + \frac {B_1^* x^2} {2!} - \frac {B_2^* x^4} {4!} + \frac {B_3^* x^6} {6!} - \cdots\) |
for $x \in \R$ such that $\size x < 2 \pi$
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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 21$: Definition of Bernoulli Numbers: $21.1$
- Weisstein, Eric W. "Bernoulli Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BernoulliNumber.html