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

${\displaystyle {\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n!}}\cdot t^{n}}$,

where cosh t is the hyperbolic cosine. The Euler numbers appear as a special value of the Euler polynomials.

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 = 1,385

E₁₀ = −50,521

E₁₂ = 2,702,765

E₁₄ = −199,360,981

E₁₆ = 19,391,512,145

E₁₈ = −2,404,879,675,441

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

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.

Explicit formulas

Iterated sum

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

$E_{2n}=i\sum _{k=1}^{2n+1} \sum _{j=0}^k {k\choose j}\frac{(-1)^j(k-2j)^{2n+1}}{2^k i^k k}$

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

Sum over partitions

The Euler number E_2n can be expressed as a sum over the even partitions of 2n,^[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 2n − 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 $K=k_{1}+\cdots +k_{n}$ 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 delta's in the above formulas restrict the sums over the k's to $2k_{1}+4k_{2}+\cdots +2nk_{n}=2n$ and to $k_{1}+3k_{2}+\cdots +(2n-1)k_{n}=2n-1$, respectively.

As an example,

$\begin{align} E_{10} & = 10! \left( - \frac{1}{10!} + \frac{2}{2!8!} + \frac{2}{4!6!} - \frac{3}{2!^2 6!}- \frac{3}{2!4!^2} +\frac{4}{2!^3 4!} - \frac{1}{2!^5}\right) \\ & = 9! \left( - \frac{1}{9!} + \frac{3}{1!^27!} + \frac{6}{1!3!5!} +\frac{1}{3!^3}- \frac{5}{1!^45!} -\frac{10}{1!^33!^2} + \frac{7}{1!^6 3!} - \frac{1}{1!^9}\right) \\ & = -50,521. \end{align}$

Determinant

E_2n 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, $A_{n}$ = $(-1)^{\frac{n-1}{2}}\frac{2^{n+1}(2^{n+1}-1)B_{n+1}}{n+1}$, where $B_{n}$ is the Bernoulli number.

For every n, ${\displaystyle {\frac {A_{n-1}}{(n-1)!}}\sin {\left({\frac {n}{2}}\pi \right)}+\sum _{m=0}^{n-1}{\frac {A_{m}}{m!(n-m-1)!}}\sin {\left({\frac {m}{2}}\pi \right)}={\frac {1}{(n-1)!}}.}$

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]

See also

• Bell number
• Bernoulli number
• Euler–Mascheroni constant

References

1. Ross Tang, "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series" Archived May 11, 2012, 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. Hassan Jolany, Roberto B. Corcino, Takao Komatsu, "More properties on multi-poly-Euler polynomials, Boletín de la Sociedad Matemática Mexicana October 2015, Volume 21, Issue 2, pp 149-162"

External links

• Hazewinkel, Michiel, ed. (2001), "Euler numbers", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Weisstein, Eric W. "Euler number". MathWorld.