Euler numbers

Euler numbers $E_n$ have the generating function ${\displaystyle \frac{1}{\cosh x}}$ such that

$$\frac{1}{\cosh x}=:\sum _{n=0}^{\mathrm{\infty }}\frac{E_n}{n!}{x}^n.$$

They are integers but have no expression for calculating them.  Their only are that the numbers with odd index (http://planetmath.org/IndexingSet) are all 0 and that

$$\text{sgn}(E_{2m})={(-1)}^m\mathit{\hspace{1em}\hspace{1em}}\text{for}\mathit{\hspace{1em}}m=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }$$

The Euler number have intimate relation to the Bernoulli numbers.  The first Euler numbers with even index are

$$E_0=1,E_2=-1,E_4=5,E_6=-61,E_8=1385,E_{10}=-50521.$$

• One can by hand determine Euler numbers by performing the division of 1 by the Taylor series of hyperbolic cosine (cf. Taylor series via division and Taylor series of hyperbolic functions).  Since  $\cosh ix=\cos x$,  the division $1:\cos x$ correspondingly gives only terms with plus sign, i.e. it shows the absolute values of the Euler numbers.
  

• The Euler numbers may also be obtained by using the Euler polynomials $E_n(x)$:

  $$E_n={\mathrm{\hspace{0.33em}2}}^n{E}_n\left(\frac{1}{2}\right)$$

• If the Euler numbers $E_k$ are denoted as symbolic powers $E^k$, then one may write the equation

  $${(E+1)}^n+{(E-1)}^n=\mathrm{\hspace{0.33em}0},$$

  which can be used as a recurrence relation for computing the values of the even-indexed Euler numbers.  Cf. the Leibniz rule for derivatives of product $fg$.

Title	Euler numbers
Canonical name	EulerNumbers
Date of creation	2014-12-02 17:43:40
Last modified on	2014-12-02 17:43:40
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	11
Author	pahio (2872)
Entry type	Definition
Classification	msc 11B68
Related topic	GudermannianFunction
Related topic	BernoulliNumber
Related topic	InverseGudermannianFunction
Related topic	HermiteNumbers