Euler number

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. 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	=	7003138500000000000♠1385
E₁₀	=	2995494790000000000♠−50521
E₁₂	=	7006270276500000000♠2702765
E₁₄	=	2991800639019000000♠−199360981
E₁₆	=	7010193915121450000♠19391512145
E₁₈	=	2987759512032455900♠−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. This article adheres to the convention adopted above.

Explicit formulas

As an iterated sum

An explicit formula for Euler numbers is:^[1]

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}}

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 $2 n$ ,^[2]

E_{2n}=(2n)!\sum _{0\leq k_{1},\ldots ,k_{n}\leq n}\left({\begin{array}{c}K\\k_{1},\ldots ,k_{n}\end{array}}\right)\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 $2 n - 1$ ,^[3]

E_{2n}=(-1)^{n-1}(2n-1)!\sum _{0\leq k_{1},\ldots ,k_{n}\leq 2n-1}\left({\begin{array}{c}K\\k_{1},\ldots ,k_{n}\end{array}}\right)\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

\left( \begin{array}{c} K \\ k_1, \ldots , k_n \end{array} \right) \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)\\&=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)\\&=-50\,521.\end{aligned}}

As a determinant

$E 2 n$ is also 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}

Asymptotic approximation

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

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

Euler zigzag numbers

The Taylor series of $sec x + tan x$ 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)!}}.

Generalized Euler numbers

Generalizations of Euler numbers include poly-Euler numbers and multi-poly-Euler numbers, which play an important role in multiple zeta functions.^[4]

References

↑ Ross Tang, "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series" Archived 2012-05-11 at the Wayback Machine.
↑ Vella, David C. (2008). "Explicit Formulas for Bernoulli and Euler Numbers". Integers. 8 (1): A1.
↑ Malenfant, J. "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers". arXiv:1103.1585.
↑ Jolany, Hassan; Corcino, Roberto B.; Komatsu, Takao (Oct 2015). "More properties on multi-poly-Euler polynomials". Boletín de la Sociedad Matemática Mexicana. 21 (2): 149–162. arXiv:1401.6271. doi:10.1007/s40590-015-0061-y.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Euler numbers", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Weisstein, Eric W. "Euler number". MathWorld.

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[1] Ross Tang, "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series" Archived 2012-05-11 at the Wayback Machine.

[2] Vella, David C. (2008). "Explicit Formulas for Bernoulli and Euler Numbers". Integers. 8 (1): A1.

[3] Malenfant, J. "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers". arXiv:1103.1585.

[4] Jolany, Hassan; Corcino, Roberto B.; Komatsu, Takao (Oct 2015). "More properties on multi-poly-Euler polynomials". Boletín de la Sociedad Matemática Mexicana. 21 (2): 149–162. arXiv:1401.6271. doi:10.1007/s40590-015-0061-y.