# Euler numbers

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

 $\frac{1}{\cosh{x}}\;=:\;\sum_{n=0}^{\infty}\frac{E_{n}}{n!}\,x^{n}.$

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

 $\mbox{sgn}(E_{2m})\;=\;(-1)^{m}\qquad\mbox{for}\quad m=0,\,1,\,2,\,\ldots$

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

 $E_{0}=1,\quad E_{2}=-1,\quad E_{4}=5,\quad E_{6}=-61,\quad E_{8}=1385,\quad E_% {10}=-50521.$
• 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 $E_{n}(x)$:

 $E_{n}\;=\;2^{n}E_{n}\!\!\left(\!\frac{1}{2}\!\right)$
• If the Euler numbers $E_{k}$ are denoted as symbolic powers $E^{k}$, then one may write the equation

 $(E\!+\!1)^{n}+(E\!-\!1)^{n}\;=\;0,$

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 EulerNumbers 2014-12-02 17:43:40 2014-12-02 17:43:40 pahio (2872) pahio (2872) 11 pahio (2872) Definition msc 11B68 GudermannianFunction BernoulliNumber InverseGudermannianFunction HermiteNumbers