Euler numbers

The odd-indexed Euler numbers are all zero. The even-indexed ones {{OEIS|id=A028296}} 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 {{OEIS|id=A000364}}. This article adheres to the convention adopted above.

The following two formulas express the Euler numbers in terms of Stirling numbers of the second kind:{{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 | s2cid= 209973489 }}{{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}}

:$E\_\{n\}=2^\{2n-1\}\backslash sum\_\{\backslash ell=1\}^\{n\}\backslash frac\{(-1)^\{\backslash ell\}S(n,\backslash ell)\}\{\backslash ell+1\}\backslash left(3\backslash left(\backslash frac\{1\}\{4\}\backslash right)^\{\backslash overline\{\backslash ell\backslash phantom\{.\}\}\}-\backslash left(\backslash frac\{3\}\{4\}\backslash right)^\{\backslash overline\{\backslash ell\backslash phantom\{.\}\}\}\backslash right),$

:$E\_\{2n\}=-4^\{2n\}\backslash sum\_\{\backslash ell=1\}^\{2n\}(-1)^\{\backslash ell\}\backslash cdot\; \backslash frac\{S(2n,\backslash ell)\}\{\backslash ell+1\}\backslash cdot\; \backslash left(\backslash frac\{3\}\{4\}\backslash right)^\{\backslash overline\{\backslash ell\backslash phantom\{.\}\}\},$

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

The Euler numbers can be defined as an recursion:

$E\_\{2n\}=-\backslash sum\_\{\{k=1\}\}^\{n\}\backslash binom\{2n\}\{2k\}E\_\{2(n-k)\},$

or alternatively:

$1=-\backslash sum\_\{\{k=1\}\}^\{n\}\backslash binom\{2n\}\{2k\}E\_\{2k\},$

Both of these recursions can be found by using the fact that.

$cos(x)sec(x)=1.$

The following two formulas express the Euler numbers as double sums{{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 }}

:$E\_\{2n\}=(2\; n+1)\backslash sum\_\{\backslash ell=1\}^\{2n\}\; (-1)^\{\backslash ell\}\backslash frac\{1\}\{2^\{\backslash ell\}(\backslash ell\; +1)\}\backslash binom\{2\; n\}\{\backslash ell\}\backslash sum\; \_\{q=0\}^\{\backslash ell\}\backslash binom\{\backslash ell\}\{q\}(2q-\backslash ell)^\{2n\},$

:$E\_\{2n\}=\backslash sum\_\{k=1\}^\{2n\}(-1)^\{k\}\; \backslash frac\{1\}\{2^\{k\}\}\backslash sum\_\{\backslash ell=0\}^\{2k\}(-1)^\{\backslash ell\; \}\; \backslash binom\{2k\}\{\backslash ell\}(k-\backslash ell)^\{2n\}.$

An explicit formula for Euler numbers is:{{cite web |url=https://oeis.org/A000111/a000111.pdf |archive-url=https://web.archive.org/web/20140409060145/http://oeis.org/A000111/a000111.pdf |archive-date=2014-04-09 |url-status=live |title=An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series |last=Tang |first=Ross |date= 2012-05-11}}

:$E\_\{2n\}=i\backslash sum\; \_\{k=1\}^\{2n+1\}\; \backslash sum\; \_\{\backslash ell=0\}^k\; \backslash binom\{k\}\{\backslash ell\}\backslash frac\{(-1)^\backslash ell(k-2\backslash ell)^\{2n+1\}\}\{2^k\; i^k\; k\},$

where {{mvar|i}} denotes the imaginary unit with {{math|i² {{=}} −1}}.

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

:$E\_\{2n\}\; =\; (2n)!\; \backslash sum\_\{0\; \backslash leq\; k\_1,\; \backslash ldots,\; k\_n\; \backslash leq\; n\}\; \backslash binom\; K\; \{k\_1,\; \backslash ldots\; ,\; k\_n\}$

\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 {{math|2n − 1}},{{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 }}

:$E\_\{2n\}\; =\; (-1)^\{n-1\}\; (2n-1)!\; \backslash sum\_\{0\; \backslash leq\; k\_1,\; \backslash ldots,\; k\_n\; \backslash leq\; 2n-1\}$

\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 {{math|K {{=}} k₁ + ··· + k_n}} and

:$\backslash binom\; K\; \{k\_1,\; \backslash ldots\; ,\; k\_n\}$

\equiv \frac{ K!}{k_1! \cdots k_n!}

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

As an example,

:$$

\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|E_2n}} is given by the determinant

:$$

\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|E_2n}} is also given by the following integrals:

:$$

\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}

W. Zhang{{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 | doi= 10.1080/00150517.1998.12428950 | url= https://www.mathstat.dal.ca/FQ/Scanned/36-2/zhang.pdf |archive-url=https://web.archive.org/web/20191123004402/https://www.mathstat.dal.ca/FQ/Scanned/36-2/zhang.pdf |archive-date=2019-11-23 |url-status=live}} obtained the following combinational identities concerning the Euler numbers. For any prime $p$, we have

:$$

(-1)^{\frac{p-1}{2}} E_{p-1} \equiv \textstyle\begin{cases} \phantom{-} 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{{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 }}

proved that, for any prime $p\; \backslash equiv\; 1\; \backslash pmod\{4\}$ and integer $\backslash alpha\backslash geq\; 1$, we have

:$E\_\{\backslash phi(p^\{\backslash alpha\})/2\}\backslash not\; \backslash equiv\; 0\; \backslash pmod\{p^\{\backslash alpha\}\},$

where $\backslash phi(n)$ is the Euler's totient function.

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

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

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

:$\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash frac\{A\_n\}\{n!\}x^n,$

where {{mvar|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, ... {{OEIS|id=A000111}}

For all even {{mvar|n}},

:$A\_n\; =\; (-1)^\backslash frac\{n\}\{2\}\; E\_n,$

where {{mvar|E_n}} is the Euler number, and for all odd {{mvar|n}},

:$A\_n\; =\; (-1)^\backslash frac\{n-1\}\{2\}\backslash frac\{2^\{n+1\}\backslash left(2^\{n+1\}-1\backslash right)B\_\{n+1\}\}\{n+1\},$

where {{mvar|B_n}} is the Bernoulli number.

For every n,

:$\backslash frac\{A\_\{n-1\}\}\{(n-1)!\}\backslash sin\{\backslash left(\backslash frac\{n\backslash pi\}\{2\}\backslash right)\}+\backslash sum\_\{m=0\}^\{n-1\}\backslash frac\{A\_m\}\{m!(n-m-1)!\}\backslash sin\{\backslash left(\backslash frac\{m\backslash pi\}\{2\}\backslash right)\}=\backslash frac\{1\}\{(n-1)!\}.${{cn|date=September 2016}}

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

{{Reflist}}

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

{{Classes of natural numbers}}

{{Leonhard Euler}}

{{DEFAULTSORT:Euler Number}}

Category:Eponymous numbers in mathematics

Category:Integer sequences

Category:Leonhard Euler