Definition:Euler Numbers

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Definition

The Euler Numbers $E_n$ are a sequence of integers defined by the exponential generating function:

$\displaystyle \operatorname {sech} x = \frac {2 e^x} {e^{2 x} + 1} = \sum_{n \mathop = 0}^\infty \frac {E_n x^n} {n!}$

where $\left\lvert{x}\right\rvert < \dfrac \pi 2$.


Sequence of Euler Numbers

The sequence of Euler numbers begins:

\(\displaystyle E_0\) \(=\) \(\displaystyle 1\)
\(\displaystyle E_2\) \(=\) \(\displaystyle -1\)
\(\displaystyle E_4\) \(=\) \(\displaystyle 5\)
\(\displaystyle E_6\) \(=\) \(\displaystyle -61\)
\(\displaystyle E_8\) \(=\) \(\displaystyle 1385\)
\(\displaystyle E_{10}\) \(=\) \(\displaystyle -50 \, 521\)
\(\displaystyle E_{12}\) \(=\) \(\displaystyle 2 \, 702 \, 765\)
\(\displaystyle E_{14}\) \(=\) \(\displaystyle -199 \, 360 \, 981\)
\(\displaystyle E_{16}\) \(=\) \(\displaystyle 19 \, 391 \, 512 \, 145\)
\(\displaystyle E_{18}\) \(=\) \(\displaystyle -2 \, 404 \, 879 \, 675 \, 441\)
\(\displaystyle E_{20}\) \(=\) \(\displaystyle 370 \, 371 \, 188 \, 237 \, 525\)
\(\displaystyle E_{22}\) \(=\) \(\displaystyle -69 \, 348 \, 874 \, 393 \, 137 \, 901\)
\(\displaystyle E_{24}\) \(=\) \(\displaystyle 15 \, 514 \, 534 \, 163 \, 557 \, 086 \, 905\)

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

\(\displaystyle \sech x\) \(=\) \(\displaystyle 1 + \sum_{n \mathop = 1}^\infty \left({-1}\right)^n \frac {E_n^* x^{2 n} } {\left({2 n}\right)!}\)
\(\displaystyle \) \(=\) \(\displaystyle 1 - \frac {E_1^* x^2} {2!} + \frac {E_2^* x^4} {4!} - \frac {E_3^* x^6} {6!} + \cdots\)

where $\left\lvert{x}\right\rvert < \dfrac \pi 2 $.


Definition 2

\(\displaystyle \sec x\) \(=\) \(\displaystyle 1 + \sum_{n \mathop = 1}^\infty \frac {E^*_n x^{2 n} } {\left({2 n}\right)!}\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + \frac {E_1^* x^2} {2!} + \frac {E_2^* x^4} {4!} + \frac {E_3^* x^6} {6!} + \cdots\)


where $\left\lvert{x}\right\rvert < \dfrac \pi 2$.


Also known as

The Euler numbers are also known as the secant numbers or zig numbers.


Also see

  • Results about Euler Numbers can be found here.


Source of Name

This entry was named for Leonhard Paul Euler.


Sources