Euler numbers explained

In mathematics, the Euler numbers are a sequence E_n of integers defined by the Taylor series expansion

(1)/(\cosh t) = (2)/(e ^{t} + e ^{-t}) = \sum ^{inf ty} _{n =0} (E _{n})/(n!) ⋅ t ^{n} ,

where

\cosh (t) is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely:E ^{nE} _{n(\tfrac} 12 ).

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 have alternating signs. Some values are:

E0	=	1
E2	=	−1
E4	=	5
E6	=	−61
E8	=
E10	=
E12	=
E14	=
E16	=
E18	=

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

Explicit formulas

In terms of Stirling numbers of the second kind

The following two formulas express the Euler numbers in terms of Stirling numbers of the second kind:^{[1] [2]}

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

where

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

As a double sum

The following two formulas express the Euler numbers as double sums^[3]

E _{2n} = ( 2 n +1 )\sum ^{2 n} _{\ell =1} ( -1 ) ^{\ell} (1)/(2 ^{\ell} (\ell +1 )) \binom{ 2 n}{\ell}\sum ^{\ell} ^{q =0} \binom{\ell}{q}( 2 q - \ell) ^{2n} , E _{2n} = \sum ^{2 n} _{k =1} ( -1 ) ^{k} (1)/(2 ^{k}) \sum ^{2 k} _{\ell =0} ( -1 ) ^{\ell} \binom{ 2 k}{\ell}(k - \ell) ^{2n} .

As an iterated sum

An explicit formula for Euler numbers is:^[4]

E _{2n} = i\sum ^{2 n +1} ^{k =1} \sum \binom{k}{\ell} ^{k} _{\ell =0} (( -1 ) ^{\ell(k-2\ell)} ^{2n+1})/(2 ^{k} i ^{k} k) ,

where denotes the imaginary unit with .

As a sum over partitions

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

E _{2n} = ( 2 n)! \sum _{0 \leq k _{1,} \ldots, k _{n} \leq n} \binom K {k _{1,} \ldots , k _{n} \delta} ^{n,\sum mk _{m}} \left( - (1)/(2 !) \right) ^{k _{1}} \left( - (1)/(4 !) \right) ^{k _{2}} … \left( - (1)/(( 2 n)!) \right) ^{k _{n}} ,

as well as a sum over the odd partitions of,^[6]

E _{2n} = ( -1 ) ^{n-1} ( 2 n -1 )! \sum _{0 \leq k _{1,} \ldots, k _{n} \leq 2 n -1} \binom K {k _{1,} \ldots , k _{n}} \delta _{2 n -1 ,\sum ( 2 m -1 )k _{m}} \left( - (1)/(1 !) \right) ^{k _{1}} \left( (1)/(3 !) \right) ^{k _{2}} … \left( (( -1 ) ^{n})/(( 2 n -1 )!) \right) ^{k _{n}} ,

where in both cases and

\binom K {k _{1,} \ldots , k _{n}} \equiv (K!)/(k _{1!} … k _{n!}) is a multinomial coefficient. The Kronecker deltas in the above formulas restrict the sums over the s to and to, respectively.

As an example,

\begin{align} E _{10} & = 10 ! \left( - (1)/(10 !) + (2)/(2 ! 8 !) + (2)/(4 ! 6 !) - (3)/(2 ! ^{2} 6 !) - (3)/(+ 2 ! 4 ! ^{2}) (4)/(2 ! ^{3} 4 !) - (1)/(2 ! ^{5}) \right) \\[ 6 pt] & = 9 ! \left( - (1)/(9 !) + (3)/(1 ! ^{27!}) + (6)/(+ 1 ! 3 ! 5 !) (1)/(3 ! ^{3}) - (5)/(- 1 ! ^{45!}) (10)/(1 ! ^{33!} ^{2}) + (7)/(1 ! ^{6} 3 !) - (1)/(1 ! ^{9}) \right) \\[ 6 pt] & = -50521 . \end{align}

As a determinant

is given by the determinant

\begin{align} E _{2n} & = ( -1 ) ^{n} ( 2 n)!~ \begin{vmatrix} (1)/(2 !) & 1 &~& ~&~\\ (1)/(4 !) & (1)/(2 !) & 1 &~&~\\ \vdots & ~ & \ddots~~ &\ddots~~ & ~\\ (1)/(( 2 n -2 )!) & (1)/(( 2 n -4 )!) & ~& (1)/(2 !) & 1 \\ (1)/(& ( 2 n)!) (1)/(( 2 n -2 )!) & … & (1)/(4 !) & (1)/(2 !) \end{vmatrix}. \end{align}

As an integral

is also given by the following integrals:

\begin{align} ( -1 ) ^{n} E _{2n} & = \int ^{inf ty} _{0} (t ^{2n})/(\cosh (\pi t)/(2)) dt = \left( (2 \pi\right))/(^{2n+1}) \int ^{inf ty} _{0} (x ^{2n})/(\cosh x) dx\\[ 8 pt] & = \left( (2 \pi\right))/(^{2n}) \int ^{1log} _{0} ^{2n} \left(\tan (\pi t)/(4) \right)dt = \left( (2 \pi\right))/(^{2n+1}) \int ^{\pi/ 2} _{0} log ^{2n} \left(\tan (x)/(2) \right)dx\\[ 8 pt] & = (2 ^{2n+3})/(\pi ^{2n+2}) \int ^{\pi/ 2} _{0} x log ^{2n} (\tan x)dx = \left( (2 \pi\right))/(^{2n+2}) \int ^{\pi} _{0} (x)/(2) log ^{2n} \left(\tan (x)/(2) \right)dx.\end{align}

Congruences

W. Zhang^[7] obtained the following combinational identities concerning the Euler numbers. For any prime

p , we have( -1 ) ^{(p -1)/(2)} E _{p-1} \equiv style\begin{cases} \phantom{ - } 0 \mod p & if p\equiv 1 \bmod 4 ; \ -2 \mod p & if p\equiv 3 \bmod 4 . \end{cases} W. Zhang and Z. Xu^[8] proved that, for any prime p \equiv 1 \pmod{ 4 } and integer \alpha\geq 1 , we haveE _{\phi(p ^{\alpha} )/ 2} \not \equiv 0 \pmod{p ^{\alpha} }, where \phi(n) is the Euler's totient function.

Lower bound

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

|E _{2} | > 8 \sqrt { (n)/(\pi) } \left( (4 n)/(\pi e) \right) ^{2} .

Euler zigzag numbers

The Taylor series of

\sec x + \tan x = \tan\left( (\pi 4)/(+) (x 2 \right))/() is\sum ^{inf ty} _{n =0} (A _{n})/(n!) x ^{n,}

where 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, ...

For all even,

A _{n} = ( -1 ) ^{(n)/(2)} E _{n,} where is the Euler number, and for all odd,A _{n} = ( -1 ) ^{(n -1)/(2)} (2 ^{n+1} \left( 2 ^{n+1} -1 \right)B _{n+1})/(n +1) , where is the Bernoulli number.

For every n,

(A _{n-1})/(\sin{\left( (n -1 )!) (n\pi)/(2) \right)} + \sum ^{n -1} _{m =0} (A _{m})/(\sin{\left( m!(n - m -1 )!) (m\pi)/(\right)} = 2) (1)/((n -1 )!) .

See also

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

Notes and References

1. Sumit Kumar . Jha . A new explicit formula for Bernoulli numbers involving the Euler number . Moscow Journal of Combinatorics and Number Theory . 8 . 4 . 385–387 . 2019 . 10.2140/moscow.2019.8.389 . 209973489 .
2. Web site: A new explicit formula for the Euler numbers in terms of the Stirling numbers of the second kind . Jha . Sumit Kumar . 15 November 2019.
3. Chun-Fu . Wei . Feng . Qi . Several closed expressions for the Euler numbers . Journal of Inequalities and Applications . 219 . 2015. 2015 . 10.1186/s13660-015-0738-9 . free .
4. Web site: An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series . https://web.archive.org/web/20140409060145/http://oeis.org/A000111/a000111.pdf . 2014-04-09 . live . Tang . Ross . 2012-05-11.
5. David C. . Vella . Explicit Formulas for Bernoulli and Euler Numbers . Integers . 8 . 1 . A1 . 2008 .
6. 1103.1585 . J. . Malenfant . Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers. math.NT . 2011 .
7. W.P.. Zhang . Some identities involving the Euler and the central factorial numbers . Fibonacci Quarterly . 36 . 4 . 154–157 . 1998 . 10.1080/00150517.1998.12428950 . https://web.archive.org/web/20191123004402/https://www.mathstat.dal.ca/FQ/Scanned/36-2/zhang.pdf . 2019-11-23 . live.
8. W.P. . Zhang . Z.F. . Xu . On a conjecture of the Euler numbers . Journal of Number Theory . 127 . 2. 283–291 . 2007 . 10.1016/j.jnt.2007.04.004 . free .