Euler number

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For other uses, see Euler number (topology) and Eulerian number. Also see e (mathematical constant),Euler number (physics) and Euler–Mascheroni constant.

In mathematics, in the area of number theory, the Euler numbers are a sequence E_n of integers defined by the following Taylor series expansion:

$\frac{1}{\cosh t} = \frac{2}{e^{t} + e^ {-t} } = \sum_{n=0}^{\infin} \frac{E_n}{n!} \cdot t^n\!$

where cosh t is the hyperbolic cosine. The Euler numbers appear as a special value of the Euler polynomials.

The odd-indexed Euler numbers are all zero. The even-indexed ones (sequence A000364 in OEIS) have alternating signs. Some values are:

E₀ = 1

E₂ = −1

E₄ = 5

E₆ = −61

E₈ = 1,385

E₁₀ = −50,521

E₁₂ = 2,702,765

E₁₄ = −199,360,981

E₁₆ = 19,391,512,145

E₁₈ = −2,404,879,675,441

Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, and/or change all signs to positive. This encyclopedia adheres to the convention adopted above.

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; see alternating permutation.

[edit] Asymptotic approximation

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

$|E_{2 n}| > 8 \sqrt { \frac{n}{\pi} } \left(\frac{4 n}{ \pi e}\right)^{2 n} \ .$

Refining this relation gives an asymptotic approximation for the Euler numbers

$|E_{2 n}| \sim 8 \sqrt { \frac{n}{\pi} } \left(\frac{4 n}{ \pi e} \cdot \frac{480 n^2 + 9}{480 n^2 -1}\right)^{2n}.$

This formula (Peter Luschny, 2007) is based on the connection of the Euler numbers with the Riemann zeta function. For example this approximation gives

$|E(1000)| \approx 3.8875618412530706152569\ldots\times 10^{2371}$

which is off only by four units in the least significant digit displayed.

[edit] Inequalities

The following two inequalities (Peter Luschny, 2007) hold for n > 4 and the arithmetic mean of the two bounds is an approximation of order n⁻³ to the absolute value of the Euler numbers E_2n.

$4 \sqrt{e}\left(\frac{4 n}{\pi e}\right)^{2n+1/2} \left[ 1+\frac{1}{24n}\right] < \left\vert E_{2n} \right\vert < 4 \sqrt{e}\left(\frac{4 n}{\pi e}\right)^{2n+1/2}\left[ 1+\frac{1}{24n} \left(1+\frac{1}{24n}\right) \right]$

Deleting the squared brackets on both sides and replacing on the right hand side the factor 4 by 5 gives simple inequalities valid for n > 0. These inequalities can be compared to related inequalities for the Bernoulli numbers.

For example for E₁₀₀₀ ×10^-2371 = 3.88756184125..., the low bound gives 3.88756182..., the high bound gives 3.88756185... and the mean value gives 3.88756184126... .

[edit] Integral representation and continuation.

The integral

$e(s) = 2e^{s i \pi/2}\int_{0}^{\infty} \frac{t^{s+1}}{e^{t\pi/2}+e^{-t\pi/2} } \frac{dt}{t}$

has as special values E_2n = e(2n) for n ≥ 0. The integral might be considered as a continuation of the Euler numbers to the complex plane and this was indeed suggested by Peter Luschny in 2004.

For example e(3) = -192β(4)Π^-4Ι and e(5) = 15360β(6)Π^-6Ι. Here β(n) denotes the Dirichlet beta function and Ι the imaginary unit.