Definition:Euler Numbers/Alternative Form

Definition

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 $\size x < \dfrac \pi 2$.

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

• Equivalence of Definitions of Alternative Form of Euler Numbers

Sources

• Weisstein, Eric W. "Euler Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerNumber.html