# 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.