Definition:Euler Numbers/Alternative Form/Definition 1
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Definition
The alternative form Euler numbers $E_n^*$ are a sequence of rational numbers defined as:
| \(\ds \sech x\) | \(=\) | \(\ds 1 + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac {E_n^* x^{2 n} } {\paren {2 n}!}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 - \frac {E_1^* x^2} {2!} + \frac {E_2^* x^4} {4!} - \frac {E_3^* x^6} {6!} + \cdots\) |
where $\size x < \dfrac \pi 2 $.
Sequence of Euler Numbers: Alternative Form
The sequence of the alternative form of Euler numbers begins:
| \(\ds E_1^*\) | \(=\) | \(\ds 1\) | \(\ds = -E_2\) | |||||||||||
| \(\ds E_2^*\) | \(=\) | \(\ds 5\) | \(\ds = E_4\) | |||||||||||
| \(\ds E_3^*\) | \(=\) | \(\ds 61\) | \(\ds = -E_6\) | |||||||||||
| \(\ds E_4^*\) | \(=\) | \(\ds 1385\) | \(\ds = E_8\) | |||||||||||
| \(\ds E_5^*\) | \(=\) | \(\ds 50 \, 521\) | \(\ds = -E_{10}\) | |||||||||||
| \(\ds E_6^*\) | \(=\) | \(\ds 2 \, 702 \, 765\) | \(\ds = E_{12}\) | |||||||||||
| \(\ds E_7^*\) | \(=\) | \(\ds 199 \, 360 \, 981\) | \(\ds = -E_{14}\) | |||||||||||
| \(\ds E_8^*\) | \(=\) | \(\ds 19 \, 391 \, 512 \, 145\) | \(\ds = E_{16}\) | |||||||||||
| \(\ds E_9^*\) | \(=\) | \(\ds 2 \, 404 \, 879 \, 675 \, 441\) | \(\ds = -E_{18}\) | |||||||||||
| \(\ds E_{10}^*\) | \(=\) | \(\ds 370 \, 371 \, 188 \, 237 \, 525\) | \(\ds = E_{20}\) | |||||||||||
| \(\ds E_{11}^*\) | \(=\) | \(\ds 69 \, 348 \, 874 \, 393 \, 137 \, 901\) | \(\ds = -E_{22}\) | |||||||||||
| \(\ds E_{12}^*\) | \(=\) | \(\ds 15 \, 514 \, 534 \, 163 \, 557 \, 086 \, 905\) | \(\ds = E_{24}\) |
where $E_2, E_4, \ldots$ are the standard form Euler numbers.
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 21$: Definition of Euler Numbers: $21.3$
- Weisstein, Eric W. "Euler Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerNumber.html