Euler numbers

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

{\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n!}}\cdot 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_{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 (sequence A028296 in the 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 (sequence A000364 in the OEIS). 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 _{\ell =1}^{n}{\frac {(-1)^{\ell }S(n,\ell )}{\ell +1}}\left(3\left({\frac {1}{4}}\right)^{\overline {\ell {\phantom {.}}}}-\left({\frac {3}{4}}\right)^{\overline {\ell {\phantom {.}}}}\right),

E_{2n}=-4^{2n}\sum _{\ell =1}^{2n}(-1)^{\ell }\cdot {\frac {S(2n,\ell )}{\ell +1}}\cdot \left({\frac {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}=(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},

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:^[4]

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 $i 2 = -1$ .

As a sum over partitions

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

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 $2 n - 1$ ,^[6]

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 $K = k 1 + \cdot\cdot\cdot + k n$ and

{\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 $k$ s to $2 k 1 + 4 k 2 + \cdot\cdot\cdot + 2 nk n = 2 n$ and to $k 1 + 3 k 2 + \cdot\cdot\cdot + (2 n - 1) k n = 2 n - 1$ , respectively.

As an example,

{\begin{aligned}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{aligned}}

As a determinant

$E 2 n$ is given by the determinant

{\begin{aligned}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{aligned}}

As an integral

$E 2 n$ is also given by the following integrals:

{\begin{aligned}(-1)^{n}E_{2n}&=\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\\[8pt]&=\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\\[8pt]&={\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^[7] 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^[8] proved that, for any prime $p\equiv 1{\pmod {4}}$ and integer $\alpha \geq 1$ , we have

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

Euler zigzag numbers

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

\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, ... (sequence A000111 in the OEIS)

For all even $n$ ,

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

where $E n$ is the Euler number, and for all odd $n$ ,

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,

{\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)!}}.

^{[citation needed]}

References

^ 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. S2CID 209973489.
^ Jha, Sumit Kumar (15 November 2019). "A new explicit formula for the Euler numbers in terms of the Stirling numbers of the second kind".
^ 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.
^ Tang, Ross (2012-05-11). "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series" (PDF). Archived (PDF) from the original on 2014-04-09.
^ Vella, David C. (2008). "Explicit Formulas for Bernoulli and Euler Numbers". Integers. 8 (1): A1.
^ Malenfant, J. (2011). "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers". arXiv:1103.1585 [math.NT].
^ Zhang, W.P. (1998). "Some identities involving the Euler and the central factorial numbers" (PDF). Fibonacci Quarterly. 36 (4): 154–157. doi:10.1080/00150517.1998.12428950. Archived (PDF) from the original on 2019-11-23.
^ 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.

External links

"Euler numbers", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Weisstein, Eric W. "Euler number". MathWorld.

[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. S2CID 209973489.

[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.

[4] Tang, Ross (2012-05-11). "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series" (PDF). Archived (PDF) from the original on 2014-04-09.

[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 [math.NT].

[7] Zhang, W.P. (1998). "Some identities involving the Euler and the central factorial numbers" (PDF). Fibonacci Quarterly. 36 (4): 154–157. doi:10.1080/00150517.1998.12428950. Archived (PDF) from the original on 2019-11-23.

[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.

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