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euler

Euler numbers and polynomials

 Calling Sequence euler(n) euler(n, x)

Parameters

 n - non-negative integer x - expression

Description

 • The function euler computes the nth Euler number, or the nth Euler polynomial in x.
 • The nth Euler polynomial E(n, x) is defined by the exponential generating function:

$\frac{2{ⅇ}^{xt}}{{ⅇ}^{t}+1}=\sum _{n=0}^{\mathrm{\infty }}\frac{E\left(n,x\right){t}^{n}}{n!}$

 • The nth Euler number E(n) is defined by the exponential generating function:

$\frac{2}{{ⅇ}^{t}+{ⅇ}^{-t}}=\sum _{n=0}^{\mathrm{\infty }}\frac{E\left(n\right){t}^{n}}{n!}$

 • The nth Euler polynomial E(n, x) and the nth Euler number E(n) are related by the following equation:

$E\left(n\right)={2}^{n}E\left(n,\frac{1}{2}\right)$

Examples

 > $\mathrm{euler}\left(6\right)$
 ${-61}$ (1)
 > $\mathrm{euler}\left(2,5\right)$
 ${20}$ (2)
 > $\mathrm{euler}\left(4,x\right)$
 ${{x}}^{{4}}{-}{2}{}{{x}}^{{3}}{+}{x}$ (3)
 > $\mathrm{euler}\left(4,\frac{1}{2}\right)$
 $\frac{{5}}{{16}}$ (4)
 > $\mathrm{euler}\left(2,5x\right)$
 ${25}{}{{x}}^{{2}}{-}{5}{}{x}$ (5)

 See Also