Euler number

:"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 OEIS|id=A000364 have alternating signs. Some values are::"E"₀ = 1 :"E"₂ = −1:"E"₄ = 5:"E"₆ = −61:"E"₈ = 1,385:"E"₁₀ = −50,521:"E"₁₂ = 2,702,765:"E"₁₄ = −199,360,981:"E"₁₆ = 19,391,512,145:"E"₁₈ = −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.

Asymptotic approximation

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

: $|E\_\{2\; n\}|\; >\; 8\; sqrt\; \{\; frac\{n\}\{pi\}\; \}\; left(frac\{4\; n\}\{\; pi\; e\}\; ight)^\{2\; n\}\; .$

Refining this relation gives an asymptotic approximation for the Euler numbers

: $|E\_\{2\; n\}|\; sim\; 8\; sqrt\; \{\; frac\{n\}\{pi\}\; \}\; left(frac\{4\; n\}\{\; pi\; e\}\; cdot\; frac\{480\; n^2\; +\; 9\}\{480\; n^2\; -1\}\; ight)^\{2n\}.$

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

: $|E(1000)|\; approx\; 3.8875618412530706152569ldots\; imes\; 10^\{2371\}$

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

Inequalities

The following two inequalities (Peter Luschny, 2007) hold for "n" > 4and the arithmetic mean of the two bounds is an
approximation of order "n"⁻³ to the absolute valueof the Euler numbers "E"_2"n".

:

Deleting the squared brackets on both sides and replacingon the right hand side the factor 4 by 5 gives simpleinequalities valid for "n" > 0. These inequalities canbe compared to related inequalities for the Bernoulli numbers.

For example for "E"₁₀₀₀ ×10^-2371 = 3.88756184125..., the low bound gives 3.88756182..., the high bound gives 3.88756185... and the mean value gives 3.88756184126... .

Integral representation and continuation.

The integral

: $e(s)\; =\; 2e^\{s\; i\; pi/2\}int\_\{0\}^\{infty\}\; frac\{t^\{s+1\{e^\{tpi/2\}+e^\{-tpi/2\}\; \}\; frac\{dt\}\{t\}$

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

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