أعداد أويلر Euler numbers

لعدم الخلط مع Eulerian number أو Eulerian number أو Euler's number.

للاستخدامات الأخرى، انظر List of things named after Leonhard Euler#Numbers

In mathematics, the Euler numbers are a sequence E_n of integers (المتتالية A122045 في OEIS) defined by the Taylor series expansion

${\displaystyle{\displaystyle{\frac{1}{\cosh t}}={\frac{2}{e^{t}+e^{-t}}}=\sum_{n=0}^{\infty}{\frac{E_{n}}{n!}}\cdot t^{n}}}$,

where ${\displaystyle{\displaystyle\cosh(t)}}$ is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely:

${\displaystyle{\displaystyle E_{n}=2^{n}E_{n}({\tfrac{1}{2}}).}}$

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 (المتتالية A028296 في OEIS) have alternating signs. Some values are:

E0	=	1
E2	=	−1
E4	=	5
E6	=	−61
E8	=	1385
E10	=	−50521
E12	=	2702765
E14	=	−199360981
E16	=	19391512145
E18	=	−2404879675441

Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive (المتتالية A000364 في OEIS). 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

${\displaystyle{\displaystyle E_{n}=2^{2n-1}\sum_{\ell=1}^{n}{\frac{(-1)^{\ell}S(n,\ell)}{\ell+1}}\left(3\left({\frac{1}{4}}\right)^{(\ell)}-\left({\frac{3}{4}}\right)^{(\ell)}\right),}}$

${\displaystyle{\displaystyle E_{2n}=-4^{2n}\sum_{\ell=1}^{2n}(-1)^{\ell}\cdot{\frac{S(2n,\ell)}{\ell+1}}\cdot\left({\frac{3}{4}}\right)^{(\ell)},}}$

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

 As a double sum

Following two formulas express the Euler numbers as double sums

${\displaystyle{\displaystyle E_{2n}=(2n+1)\sum_{\ell=1}^{2n}(-1)^{\ell}{\frac{1}{2^{\ell}(\ell+1)}}{\binom{2n}{\ell}}\sum_{q=0}^{\ell}{\binom{\ell}{q}}(2q-\ell)^{2n},}}$

${\displaystyle{\displaystyle E_{2n}=\sum_{k=1}^{2n}(-1)^{k}{\frac{1}{2^{k}}}\sum_{\ell=0}^{2k}(-1)^{\ell}{\binom{2k}{\ell}}(k-\ell)^{2n}.}}$

 As an iterated sum

An explicit formula for Euler numbers is:

${\displaystyle{\displaystyle E_{2n}=i\sum_{k=1}^{2n+1}\sum_{\ell=0}^{k}{\binom{k}{\ell}}{\frac{(-1)^{\ell}(k-2\ell)^{2n+1}}{2^{k}i^{k}k}},}}$

where i denotes the imaginary unit with i² = −1.

 As a sum over partitions

The Euler number E_2n can be expressed as a sum over the even partitions of 2n,

${\displaystyle{\displaystyle E_{2n}=(2n)!\sum_{0\leq k_{1},\ldots,k_{n}\leq n}{\binom{K}{k_{1},\ldots,k_{n}}}\delta_{n,\sum mk_{m}}\left(-{\frac{1}{2!}}\right)^{k_{1}}\left(-{\frac{1}{4!}}\right)^{k_{2}}\cdots\left(-{\frac{1}{(2n)!}}\right)^{k_{n}},}}$

as well as a sum over the odd partitions of 2n − 1,

${\displaystyle{\displaystyle E_{2n}=(-1)^{n-1}(2n-1)!\sum_{0\leq k_{1},\ldots,k_{n}\leq 2n-1}{\binom{K}{k_{1},\ldots,k_{n}}}\delta_{2n-1,\sum(2m-1)k_{m}}\left(-{\frac{1}{1!}}\right)^{k_{1}}\left({\frac{1}{3!}}\right)^{k_{2}}\cdots\left({\frac{(-1)^{n}}{(2n-1)!}}\right)^{k_{n}},}}$

where in both cases K = k₁ + ··· + k_n and

${\displaystyle{\displaystyle{\binom{K}{k_{1},\ldots,k_{n}}}\equiv{\frac{K!}{k_{1}!\cdots k_{n}!}}}}$

is a multinomial coefficient. The Kronecker deltas in the above formulas restrict the sums over the ks to 2k₁ + 4k₂ + ··· + 2nk_n = 2n and to k₁ + 3k₂ + ··· + (2n − 1)k_n = 2n − 1, respectively.

As an example,

${\displaystyle{\displaystyle{\begin{aligned} \displaystyle E_{10}&\displaystyle=10!\left(-{\frac{1}{10!}}+{\frac{2}{2!\,8!}}+{\frac{2}{4!\,6!}}-{\frac{3}{2!^{2}\,6!}}-{\frac{3}{2!\,4!^{2}}}+{\frac{4}{2!^{3}\,4!}}-{\frac{1}{2!^{5}}}\right)\\ &\displaystyle=9!\left(-{\frac{1}{9!}}+{\frac{3}{1!^{2}\,7!}}+{\frac{6}{1!\,3!\,5!}}+{\frac{1}{3!^{3}}}-{\frac{5}{1!^{4}\,5!}}-{\frac{10}{1!^{3}\,3!^{2}}}+{\frac{7}{1!^{6}\,3!}}-{\frac{1}{1!^{9}}}\right)\\ &\displaystyle=-50\,521.\end{aligned}}}}$

 As a determinant

E_2n is given by the determinant

${\displaystyle{\displaystyle{\begin{aligned} \displaystyle E_{2n}&\displaystyle=(-1)^{n}(2n)!~{}{\begin{vmatrix}{\frac{1}{2!}}&1&&&\\ {\frac{1}{4!}}&{\frac{1}{2!}}&1&&\\ \vdots&&\ddots&\ddots&\\ {\frac{1}{(2n-2)!}}&{\frac{1}{(2n-4)!}}&&{\frac{1}{2!}}&1\\ {\frac{1}{(2n)!}}&{\frac{1}{(2n-2)!}}&\cdots&{\frac{1}{4!}}&{\frac{1}{2!}}\end{vmatrix}}.\end{aligned}}}}$

 As an integral

E_2n is also given by the following integrals:

${\displaystyle{\displaystyle{\begin{aligned} \displaystyle(-1)^{n}E_{2n}&\displaystyle=\int_{0}^{\infty}{\frac{t^{2n}}{\cosh{\frac{\pi t}{2}}}}\;dt=\left({\frac{2}{\pi}}\right)^{2n+1}\int_{0}^{\infty}{\frac{x^{2n}}{\cosh x}}\;dx\\ &\displaystyle=\left({\frac{2}{\pi}}\right)^{2n}\int_{0}^{1}\log^{2n}\left(\tan{\frac{\pi t}{4}}\right)\,dt=\left({\frac{2}{\pi}}\right)^{2n+1}\int_{0}^{\pi/2}\log^{2n}\left(\tan{\frac{x}{2}}\right)\,dx\\ &\displaystyle={\frac{2^{2n+3}}{\pi^{2n+2}}}\int_{0}^{\pi/2}x\log^{2n}(\tan x)\,dx=\left({\frac{2}{\pi}}\right)^{2n+2}\int_{0}^{\pi}{\frac{x}{2}}\log^{2n}\left(\tan{\frac{x}{2}}\right)\,dx.\end{aligned}}}}$

 Congruences

W. Zhang obtained the following combinational identities concerning the Euler numbers, for any prime ${\displaystyle{\displaystyle p}}$, we have

${\displaystyle{\displaystyle(-1)^{\frac{p-1}{2}}E_{p-1}\equiv\textstyle{\begin{cases}0\mod p&{\text{if }}p\equiv 1{\bmod{4}};\\ -2\mod p&{\text{if }}p\equiv 3{\bmod{4}}.\end{cases}}}}$

W. Zhang and Z. Xu proved that, for any prime ${\displaystyle{\displaystyle p\equiv 1{\pmod{4}}}}$ and integer ${\displaystyle{\displaystyle\alpha\geq 1}}$, we have

${\displaystyle{\displaystyle E_{\phi(p^{\alpha})/2}\not\equiv 0{\pmod{p^{\alpha}}}}}$

where ${\displaystyle{\displaystyle\phi(n)}}$ is the Euler's totient function.

 Asymptotic approximation

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

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

 Euler zigzag numbers

The Taylor series of ${\displaystyle{\displaystyle\sec x+\tan x=\tan\left({\frac{\pi}{4}}+{\frac{x}{2}}\right)}}$ is

${\displaystyle{\displaystyle\sum_{n=0}^{\infty}{\frac{A_{n}}{n!}}x^{n},}}$

where A_n 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, ... (المتتالية A000111 في OEIS)

For all even n,

${\displaystyle{\displaystyle A_{n}=(-1)^{\frac{n}{2}}E_{n},}}$

where E_n is the Euler number; and for all odd n,

${\displaystyle{\displaystyle A_{n}=(-1)^{\frac{n-1}{2}}{\frac{2^{n+1}\left(2^{n+1}-1\right)B_{n+1}}{n+1}},}}$

where B_n is the Bernoulli number.

For every n,

${\displaystyle{\displaystyle{\frac{A_{n-1}}{(n-1)!}}\sin{\left({\frac{n\pi}{2}}\right)}+\sum_{m=0}^{n-1}{\frac{A_{m}}{m!(n-m-1)!}}\sin{\left({\frac{m\pi}{2}}\right)}={\frac{1}{(n-1)!}}.}}$^{[بحاجة لمصدر]}

 انظر أيضا

• Bell number
• عدد برنولي
• Dirichlet beta function
• Euler–Mascheroni constant

 وصلات خارجية

• Hazewinkel, Michiel, ed. (2001), "Euler numbers", Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
• Eric W. Weisstein, Euler number at MathWorld.