Euler numbers

In mathematics, the Euler numbers are a sequence ''E_n'' of integers defined by the Taylor series expansion

:$\frac = \frac = \sum_^\infty \frac \cdot t^n$,

where $\cosh (t)$ is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely:
:$E_n=2^nE_n(\tfrac 12).$

The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.

Examples

The odd-indexed Euler numbers are all zero. The even-indexed ones have alternating signs. Some values are:
:
Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive . This article adheres to the convention adopted above.

Explicit formulas

In terms of Stirling numbers of the second kind

Following two formulas express the Euler numbers in terms of Stirling numbers of the second kind

:$E_=2^\sum_^\frac\left(3\left(\frac\right)^-\left(\frac\right)^\right),$
:$E_=-4^\sum_^(-1)^\cdot \frac\cdot \left(\frac\right)^,$

where $S(n,\ell)$ denotes the Stirling numbers of the second kind, and $x^=(x)(x+1)\cdots (x+\ell-1)$ denotes the rising factorial.

As a double sum

Following two formulas express the Euler numbers as double sums
:$E_=(2 n+1)\sum_^ (-1)^\frac\binom\sum _^\binom(2q-\ell)^,$
:$E_=\sum_^(-1)^ \frac\sum_^(-1)^ \binom(k-\ell)^.$

As an iterated sum

An explicit formula for Euler numbers is:

:$E_=i\sum _^ \sum _^k \binom\frac,$

where denotes the imaginary unit with .

As a sum over partitions

The Euler number can be expressed as a sum over the even partitions of ,

:$E_ = (2n)! \sum_ \binom K \delta_ \left( -\frac \right)^ \left( -\frac \right)^ \cdots \left( -\frac \right)^ ,$

as well as a sum over the odd partitions of ,

:$E_ = (-1)^ (2n-1)! \sum_ \binom K \delta_ \left( -\frac \right)^ \left( \frac \right)^ \cdots \left( \frac \right)^ ,$

where in both cases and
:$\binom K \equiv \frac$
is a multinomial coefficient. The Kronecker deltas in the above formulas restrict the sums over the s to and to , respectively.

As an example,
:
\begin
E_ & = 10! \left( - \frac + \frac + \frac
- \frac- \frac +\frac - \frac\right) \\ pt& = 9! \left( - \frac + \frac + \frac
+\frac- \frac -\frac + \frac - \frac\right) \\ pt& = -50\,521.
\end

As a determinant

is given by the determinant

:$\begin E_ &=(-1)^n (2n)!~ \begin \frac& 1 &~& ~&~\\ \frac& \frac & 1 &~&~\\ \vdots & ~ & \ddots~~ &\ddots~~ & ~\\ \frac& \frac& ~&\frac & 1\\ \frac&\frac& \cdots & \frac & \frac\end. \end$

As an integral

is also given by the following integrals:
:
\begin
(-1)^n E_ & = \int_0^\infty \frac\; dt =\left(\frac2\pi\right)^ \int_0^\infty \frac\; dx\\ pt&=\left(\frac2\pi\right)^ \int_0^1\log^\left(\tan \frac \right)\,dt =\left(\frac2\pi\right)^\int_0^ \log^\left(\tan \frac \right)\,dx\\ pt&= \frac \int_0^ x \log^ (\tan x)\,dx = \left(\frac2\pi\right)^ \int_0^\pi \frac \log^ \left(\tan \frac \right)\,dx.\end

Congruences

W. Zhang obtained the following combinational identities concerning the Euler numbers, for any prime $p$, we have
:$(-1)^ E_ \equiv \textstyle\begin 0 \mod p &\textp\equiv 1\bmod 4; \\ -2 \mod p & \textp\equiv 3\bmod 4. \end$
W. Zhang and Z. Xu
proved that, for any prime $p \equiv 1 \pmod$ and integer $\alpha\geq 1$, we have
:$E_\not \equiv 0 \pmod$
where $\phi(n)$ is the Euler's totient function.

Asymptotic approximation

The Euler numbers grow quite rapidly for large indices as
they have the following lower bound

: $, E_, > 8 \sqrt \left(\frac\right)^.$

Euler zigzag numbers

The Taylor series of $\sec x + \tan x = \tan\left(\frac\pi4 + \frac x2\right)$ is

:$\sum_^ \fracx^n,$

where is the Euler zigzag numbers, beginning with
:1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, ...

For all even ,
:$A_n = (-1)^\frac E_n,$
where is the Euler number; and for all odd ,
:$A_n = (-1)^\frac\frac,$
where is the Bernoulli number.

For every ''n'',
:$\frac\sin+\sum_^\frac\sin=\frac.$

See also

* Bell number
* Bernoulli number
* Dirichlet beta function
* Euler–Mascheroni constant

References

External links

*
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{{DEFAULTSORT:Euler Number
Integer sequences
Leonhard Euler