Definition:Euler Numbers
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Definition
The Euler Numbers $E_n$ are a sequence of integers defined by the exponential generating function:
- $\ds \sech x = \frac {2 e^x} {e^{2 x} + 1} = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}$
where $\size x < \dfrac \pi 2$.
Recurrence Relation
- $E_n = \begin {cases} 1 & : n = 0 \\ \\ 0 & : n > 0, n = 2 r + 1 \\ \\ \ds -\sum_{k \mathop = 0}^{n - 1} \binom {2 n} {2 k} E_{2 k} & : n > 0, n = 2 r \end {cases}$
Sequence of Euler Numbers
The sequence of Euler numbers begins:
| \(\ds E_0\) | \(=\) | \(\ds 1\) | ||||||||||||
| \(\ds E_2\) | \(=\) | \(\ds -1\) | ||||||||||||
| \(\ds E_4\) | \(=\) | \(\ds 5\) | ||||||||||||
| \(\ds E_6\) | \(=\) | \(\ds -61\) | ||||||||||||
| \(\ds E_8\) | \(=\) | \(\ds 1385\) | ||||||||||||
| \(\ds E_{10}\) | \(=\) | \(\ds -50 \, 521\) | ||||||||||||
| \(\ds E_{12}\) | \(=\) | \(\ds 2 \, 702 \, 765\) | ||||||||||||
| \(\ds E_{14}\) | \(=\) | \(\ds -199 \, 360 \, 981\) | ||||||||||||
| \(\ds E_{16}\) | \(=\) | \(\ds 19 \, 391 \, 512 \, 145\) | ||||||||||||
| \(\ds E_{18}\) | \(=\) | \(\ds -2 \, 404 \, 879 \, 675 \, 441\) | ||||||||||||
| \(\ds E_{20}\) | \(=\) | \(\ds 370 \, 371 \, 188 \, 237 \, 525\) | ||||||||||||
| \(\ds E_{22}\) | \(=\) | \(\ds -69 \, 348 \, 874 \, 393 \, 137 \, 901\) | ||||||||||||
| \(\ds E_{24}\) | \(=\) | \(\ds 15 \, 514 \, 534 \, 163 \, 557 \, 086 \, 905\) |
Odd index Euler numbers are all $0$.
Alternative Form
An alternative form of the Euler numbers can often be found.
Usually denoted with the symbol ${E_n}^*$, they are generally not used on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Definition 1
| \(\ds \sech x\) | \(=\) | \(\ds 1 + \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac { {E_n}^* x^{2 n} } {\paren {2 n}!}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 - \frac { {E_1}^* x^2} {2!} + \frac { {E_2}^* x^4} {4!} - \frac { {E_3}^* x^6} {6!} + \cdots\) |
where $\size x < \dfrac \pi 2 $.
Definition 2
| \(\ds \sec x\) | \(=\) | \(\ds 1 + \sum_{n \mathop = 1}^\infty \frac { {E_n}^* x^{2 n} } {\paren {2 n}!}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 + \frac { {E_1}^* x^2} {2!} + \frac { {E_2}^* x^4} {4!} + \frac { {E_3}^* x^6} {6!} + \cdots\) |
where $\size x < \dfrac \pi 2$.
Also known as
The Euler numbers are also known as the secant numbers or zig numbers.
Also see
- Results about the Euler numbers can be found here.
Source of Name
This entry was named for Leonhard Paul Euler.
Sources
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Euler numbers
- Weisstein, Eric W. "Euler Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerNumber.html