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{{Use American English|date = March 2019}}
{{short description|Integers occurring in the coefficients of the Taylor series of 1/cosh t}}
{{confused|Eulerian number|Euler's number}}
{{other uses|List of things named after Leonhard Euler#Euler's numbers}}
In [[mathematics]], the '''Euler numbers''' are a [[sequence]] ''E<sub>n</sub>'' of [[integer]]s {{OEIS|A122045}} defined by the [[Taylor series]] expansion

:<math>\frac{1}{\cosh t} = \frac{2}{e^{t} + e^ {-t} } = \sum_{n=0}^\infty \frac{E_n}{n!} \cdot t^n</math>,

where {{math|cosh ''t''}} is the [[Hyperbolic function|hyperbolic cosine]]. The Euler numbers are related to a special value of the [[Euler polynomials]], namely:
:<math>E_n=2^nE_n(\tfrac 12).</math>

The Euler numbers appear in the [[Taylor series]] expansions of the [[Trigonometric functions|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 permutation]]s of a set with an even number of elements.

== Examples ==
The odd-indexed Euler numbers are all [[0 (number)|zero]]. The even-indexed ones {{OEIS|id=A028296}} have alternating signs. Some values are:
:{|
|''E''<sub>0</sub> ||=||align=right| 1
|-
|''E''<sub>2</sub> ||=||align=right| −1
|-
|''E''<sub>4</sub> ||=||align=right| 5
|-
|''E''<sub>6</sub> ||=||align=right| −61
|-
|''E''<sub>8</sub> ||=||align=right| {{val|1385|fmt=gaps}}
|-
|''E''<sub>10</sub> ||=||align=right| {{val|−50521}}
|-
|''E''<sub>12</sub> ||=||align=right| {{val|2,702,765}}
|-
|''E''<sub>14</sub> ||=||align=right| {{val|−199,360,981}}
|-
|''E''<sub>16</sub> ||=||align=right| {{val|19,391,512,145}}
|-
|''E''<sub>18</sub> ||=||align=right| {{val|−2,404,879,675,441}}
|}
Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive {{OEIS|id=A000364}}. 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]]<ref>{{cite journal | first1=Sumit Kumar | last1= Jha | title=A new explicit formula for Bernoulli numbers involving the Euler number | journal=Moscow Journal of Combinatorics and Number Theory | volume=8 | issue=4 | pages=385–387 | year=2019 | url= https://projecteuclid.org/euclid.moscow/1572314455| doi= 10.2140/moscow.2019.8.389 }}</ref>
<ref>{{cite web |url=https://osf.io/smw7h/ |title=A new explicit formula for the Euler numbers in terms of the Stirling numbers of the second kind |last=Jha |first=Sumit Kumar |date= 15 November 2019}}</ref>

:<math> E_{r}=2^{2r-1}\sum_{k=1}^{r}\frac{(-1)^{k}S(r,k)}{k+1}\left(3\left(\frac{1}{4}\right)^{(k)}-\left(\frac{3}{4}\right)^{(k)}\right), </math>
:<math> E_{2l}=-4^{2l}\sum_{k=1}^{2l}(-1)^{k}\cdot \frac{S(2l,k)}{k+1}\cdot \left(\frac{3}{4}\right)^{(k)},</math>

where <math> S(r,k) </math> denotes the [[Stirling numbers of the second kind]], and <math> x^{(n)}=(x)(x+1)\cdots (x+n-1) </math> denotes the [[Falling and rising factorials|rising factorial]].

===As a double sum===
Following two formulas express the Euler numbers as double sums<ref>{{cite journal | first1=Chun-Fu | last1= Wei | first2=Feng | last2=Qi | title=Several closed expressions for the Euler numbers | journal=Journal of Inequalities and Applications | volume=219 | issue=2015| year=2015 | doi= 10.1186/s13660-015-0738-9 | doi-access=free }}
</ref>
:<math>E_{2k}=(2 k+1)\sum_{\ell=1}^{2k} (-1)^{\ell}\frac{1}{2^{\ell}(\ell +1)}\binom{2 k}{\ell}\sum _{q=0}^{\ell}\binom{\ell}{q}(2q-\ell)^{2k}, </math>
:<math>E_{2k}=\sum_{i=1}^{2k}(-1)^{i} \frac{1}{2^{i}}\sum_{\ell=0}^{2i}(-1)^{\ell } \binom{2i}{\ell}(i-\ell)^{2k}. </math>

===As an iterated sum===
An explicit formula for Euler numbers is:<ref>{{cite web |url=https://oeis.org/A000111/a000111.pdf |title=An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series |last=Tang |first=Ross |date= 2012-05-11}}
</ref>

:<math>E_{2n}=i\sum _{k=1}^{2n+1} \sum _{j=0}^k \binom{k}{j}\frac{(-1)^j(k-2j)^{2n+1}}{2^k i^k k},</math>

where {{mvar|i}} denotes the [[imaginary unit]] with {{math|''i''<sup>2</sup> {{=}} −1}}.

===As a sum over partitions===
The Euler number {{math|''E''<sub>2''n''</sub>}} can be expressed as a sum over the even [[Partition (number theory)|partitions]] of {{math|2''n''}},<ref>{{cite journal | first1=David C. | last1= Vella | title=Explicit Formulas for Bernoulli and Euler Numbers | journal=Integers | volume=8 | issue=1 | pages=A1 | year=2008 | url= http://www.integers-ejcnt.org/vol8.html}}</ref>

:<math> 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} ,</math>

as well as a sum over the odd partitions of {{math|2''n'' − 1}},<ref>{{cite arxiv | eprint=1103.1585 | first1= J. | last1=Malenfant | title=Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers| class= math.NT | year= 2011 }}</ref>

:<math> 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} , </math>

where in both cases {{math|''K'' {{=}} ''k''<sub>1</sub> + ··· + ''k<sub>n</sub>''}} and
:<math> \binom K {k_1, \ldots , k_n}
\equiv \frac{ K!}{k_1! \cdots k_n!}</math>
is a [[multinomial coefficient]]. The [[Kronecker delta]]s in the above formulas restrict the sums over the {{mvar|k}}s to {{math|2''k''<sub>1</sub> + 4''k''<sub>2</sub> + ··· + 2''nk<sub>n</sub>'' {{=}} 2''n''}} and to {{math|''k''<sub>1</sub> + 3''k''<sub>2</sub> + ··· + (2''n'' − 1)''k<sub>n</sub>'' {{=}} 2''n'' − 1}}, respectively.

As an example,
:<math>
\begin{align}
E_{10} & = 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) \\[6pt]
& = 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) \\[6pt]
& = -50\,521.
\end{align}
</math>

===As a determinant===
{{math|''E''<sub>2''n''</sub>}} is given by the [[determinant]]

:<math>
\begin{align}
E_{2n} &=(-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{align}
</math>

===As an integral===
{{math|''E''<sub>2''n''</sub>}} is also given by the following integrals:
:<math>
\begin{align}
(-1)^n E_{2n} & = \int_0^\infty \frac{t^{2n}}{\cosh\frac{\pi t}2}\; dt =\left(\frac2\pi\right)^{2n+1} \int_0^\infty \frac{x^{2n}}{\cosh x}\; dx\\[8pt]
&=\left(\frac2\pi\right)^{2n} \int_0^1\log^{2n}\left(\tan \frac{\pi t}{4} \right)\,dt =\left(\frac2\pi\right)^{2n+1}\int_0^{\pi/2} \log^{2n}\left(\tan \frac{x}{2} \right)\,dx\\[8pt]
&= \frac{2^{2n+3}}{\pi^{2n+2}} \int_0^{\pi/2} x \log^{2n} (\tan x)\,dx = \left(\frac2\pi\right)^{2n+2} \int_0^\pi \frac{x}{2} \log^{2n} \left(\tan \frac{x}{2} \right)\,dx.\end{align}
</math>

==Congruences==
W. Zhang<ref>{{cite journal | first1=W.P.| last1= Zhang | title=Some identities involving the Euler and the central factorial numbers | journal=Fibonacci Quarterly | volume=36 | issue=4 | pages=154–157 | year=1998 | url= https://www.mathstat.dal.ca/FQ/Scanned/36-2/zhang.pdf}}</ref> obtained the following combinational identities concerning the Euler numbers, for any prime <math> p </math>, we have
:<math>
(-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}
</math>
W. Zhang and Z. Xu<ref>{{cite journal | first1=W.P. | last1= Zhang | first2= Z.F. | last2=Xu | title=On a conjecture of the Euler numbers | journal=Journal of Number Theory | volume=127 | issue=2| pages= 283–291 | year=2007 | doi= 10.1016/j.jnt.2007.04.004 | doi-access=free }}
</ref> proved that, for any prime <math> p \equiv 1 \pmod{4} </math> and integer <math> \alpha\geq 1 </math>, we have
:<math> E_{\phi(p^{\alpha})/2}\not \equiv 0 \pmod{p^{\alpha}} </math>
where <math>\phi(n)</math> is the [[Euler's totient function]].

==Asymptotic approximation==

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

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

==Euler zigzag numbers==
The [[Taylor series]] of <math>\sec x + \tan x = \tan\left(\frac\pi4 + \frac x2\right)</math> is

:<math>\sum_{n=0}^{\infty} \frac{A_n}{n!}x^n,</math>

where {{mvar|A<sub>n</sub>}} is the [[Alternating permutation|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, ... {{OEIS|id=A000111}}

For all even {{mvar|n}},
:<math>A_n = (-1)^\frac{n}{2} E_n,</math>
where {{mvar|''E<sub>n</sub>''}} is the Euler number; and for all odd {{mvar|n}},
:<math>A_n = (-1)^\frac{n-1}{2}\frac{2^{n+1}\left(2^{n+1}-1\right)B_{n+1}}{n+1},</math>
where {{mvar|''B<sub>n</sub>''}} is the [[Bernoulli number]].

For every ''n'',
:<math>\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)!}.</math>{{cn|date=September 2016}}

==See also==
* [[Bell number]]
* [[Bernoulli number]]
* [[Euler–Mascheroni constant]]

==References==
{{Reflist}}
{{Leonhard Euler}}

==External links==
* {{springer|title=Euler numbers|id=p/e036540}}
* {{MathWorld|urlname=EulerNumber|title=Euler number}}

{{Classes of natural numbers}}

{{DEFAULTSORT:Euler Number}}
[[Category:Integer sequences]]
[[Category:Leonhard Euler]]

Sayfanın 11.58, 13 Temmuz 2021 tarihindeki hâli

Şablon:Use American English

Şablon:Confused Şablon:Other uses In mathematics, the Euler numbers are a sequence En of integers (OEIS'de A122045 dizisi) defined by the Taylor series expansion

,

where cosh t is the hyperbolic cosine. The Euler numbers are related to a special value of the Euler polynomials, namely:

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 (OEIS'de A028296 dizisi) have alternating signs. Some values are:

E0 = 1
E2 = −1
E4 = 5
E6 = −61
E8 = 1385
E10 = Error in {{değer}}: parametre 1 geçerli bir sayı değil.
E12 = Error in {{değer}}: parametre 1 geçerli bir sayı değil.
E14 = Error in {{değer}}: parametre 1 geçerli bir sayı değil.
E16 = Error in {{değer}}: parametre 1 geçerli bir sayı değil.
E18 = Error in {{değer}}: parametre 1 geçerli bir sayı değil.

Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive (OEIS'de A000364 dizisi). 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[1] [2]

where denotes the Stirling numbers of the second kind, and denotes the rising factorial.

As a double sum

Following two formulas express the Euler numbers as double sums[3]

As an iterated sum

An explicit formula for Euler numbers is:[4]

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

As a sum over partitions

The Euler number E2n can be expressed as a sum over the even partitions of 2n,[5]

as well as a sum over the odd partitions of 2n − 1,[6]

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

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

As an example,

As a determinant

E2n is given by the determinant

As an integral

E2n is also given by the following integrals:

Congruences

W. Zhang[7] obtained the following combinational identities concerning the Euler numbers, for any prime , we have

W. Zhang and Z. Xu[8] proved that, for any prime and integer , we have

where is the Euler's totient function.

Asymptotic approximation

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

Euler zigzag numbers

The Taylor series of is

where An 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, ... (OEIS'de A000111 dizisi)

For all even n,

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

where Bn is the Bernoulli number.

For every n,

[kaynak belirtilmeli]

See also

References

  1. ^ Jha, Sumit Kumar (2019). "A new explicit formula for Bernoulli numbers involving the Euler number". Moscow Journal of Combinatorics and Number Theory. 8 (4): 385–387. doi:10.2140/moscow.2019.8.389. 
  2. ^ Jha, Sumit Kumar (15 November 2019). "A new explicit formula for the Euler numbers in terms of the Stirling numbers of the second kind". 
  3. ^ Wei, Chun-Fu; Qi, Feng (2015). "Several closed expressions for the Euler numbers". Journal of Inequalities and Applications. 219 (2015). doi:10.1186/s13660-015-0738-9.  Geçersiz |doi-access=free (yardım)
  4. ^ Tang, Ross (2012-05-11). "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series" (PDF). 
  5. ^ Vella, David C. (2008). "Explicit Formulas for Bernoulli and Euler Numbers". Integers. 8 (1): A1. 
  6. ^ Malenfant, J. (2011). "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers". arXiv:1103.1585 $2. 
  7. ^ Zhang, W.P. (1998). "Some identities involving the Euler and the central factorial numbers" (PDF). Fibonacci Quarterly. 36 (4): 154–157. 
  8. ^ Zhang, W.P.; Xu, Z.F. (2007). "On a conjecture of the Euler numbers". Journal of Number Theory. 127 (2): 283–291. doi:10.1016/j.jnt.2007.04.004.  Geçersiz |doi-access=free (yardım)

External links

Şablon:Classes of natural numbers