Euler's numbers

This article covers a sequence of integers. For other numbers and number sequences named after Euler, see Euler's numbers (explanation of terms) .

The Euler numbers or sometimes Euler numbers (by Leonhard Euler are) a sequence of integers, by the Taylor expansion of the hyperbolic secant hyperbolic ${\displaystyle \,E_{n}}$

${\displaystyle \operatorname {sech} (x)={\frac {1}{\cosh(x)}}={\frac {2}{e^{x}+e^{-x}}}=\sum _{n=0}^{\infty }E_{n}{\frac {x^{n}}{n!}}}$

are defined. They are not to be confused with the two-parameter Euler numbers E (n, k).

Numerical values

The first Euler's numbers ≠ 0 are ${\displaystyle E_{n}}$

${\displaystyle n}$	${\displaystyle E_{n}}$
0	1
2	−1
4th	5
6th	−61
8th	1385
10	−50521
12	2702765
14th	−199360981
16	19391512145
18th	−2404879675441
20th	370371188237525

All Euler's numbers with an odd index are zero, while those with an even index have alternating signs . Furthermore, with the exception of E _0, when dividing by 10 , the positive values ​​have the remainder 5 and the negative values modulo 10 have the remainder −1 or value 9.

Some authors leave out the numbers with an odd index altogether, halve the indices, so to speak, since the values ​​with 0 are not considered there, and define their Euler numbers as the remaining sequence. Sometimes Euler's numbers are also defined in such a way that they are all positive, i.e. correspond to ours . ${\displaystyle (-1)^{n}E_{2n}}$

properties

Asymptotic behavior

The following applies to the asymptotic behavior of Euler's numbers

${\displaystyle E_{2n}\sim (-1)^{n}\,8\,{\sqrt {\frac {n}{\pi }}}\left({\frac {4n}{\pi e}}\right)^{2n}=(-1)^{n}{\frac {\sqrt {e}}{2}}\left({\frac {4n}{\pi e}}\right)^{2n+{\frac {1}{2}}}}$

or more precisely

${\displaystyle {\frac {E_{2n}}{(2n)!}}\sim 2\,(-1)^{n}\left({\frac {2}{\pi }}\right)^{2n+1}}$

with the ~ -equivalence notation.

Recursion equation

An easy-to-remember form of the recursion equation with the seed is ${\displaystyle E_{0}=1}$

${\displaystyle \forall \,n\in \mathbb {N} \colon \quad (E+1)^{n}+(E-1)^{n}=0}$

where is to be interpreted as and from what ${\displaystyle E^{n}}$ ${\displaystyle E_{n}}$

${\displaystyle \forall \,n\in \mathbb {N} \colon \quad \sum _{k=0}^{n}\left(1+(-1)^{n-k}\right){n \choose k}E_{k}=0}$

or the explicit form through index transformation

${\displaystyle \forall \,n\in \mathbb {N} \colon \quad E_{n}=-\sum _{k=1}^{\lfloor n/2\rfloor }{n \choose 2k}E_{n-2k}}$

follows.

Closed representations

The Euler's numbers can even be made exact

${\displaystyle \forall \,n\in \mathbb {N} _{0}\colon \quad E_{2n}={\frac {(2n)!\,2^{2n+2}}{(-1)^{n}\pi ^{2n+1}}}\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)^{2n+1}}}={\frac {(2n)!\,2}{(-1)^{n}(2\pi )^{2n+1}}}\left(\zeta (2n+1,{\tfrac {1}{4}})-\zeta (2n+1,{\tfrac {3}{4}})\right)}$

using the Hurwitz zeta function if is, represent. And using its functional equation (there with m = 1, n = 4) the elegant relationship ${\displaystyle \zeta }$ ${\displaystyle n\not =0}$

${\displaystyle E_{2n}=4^{2n+1}\zeta (-2n,{\tfrac {1}{4}})}$

which identifies these numbers as scaled function values ​​of this holomorphic function . So we also get ${\displaystyle \mathbb {C} \setminus \{1\}}$

${\displaystyle E_{2n}=-4^{2n+1}{\frac {B_{2n+1}({\tfrac {1}{4}})}{2n+1}}}$

which creates a direct connection with the Bernoulli polynomials and thus with the Bernoulli numbers . Also applies ${\displaystyle B_{n}(x)}$

${\displaystyle E_{2n}={\frac {(-1)^{n}4^{n+1}(2n)!}{\pi ^{2n+1}}}\cdot \beta (2n+1),}$

where the Dirichlet beta function denotes. ${\displaystyle \beta (s)}$

Euler's polynomials

Not to be confused with the Euler polynomials

Euler's polynomials are mostly characterized by their generating function ${\displaystyle {\text{E}}_{n}\colon \mathbb {R} \to \mathbb {R} }$

${\displaystyle {\frac {2e^{xt}}{e^{t}+1}}=\sum _{n=0}^{\infty }{\text{E}}_{n}(x){\frac {t^{n}}{n!}}}$

implicitly defined. The first are

${\displaystyle {\text{E}}_{0}(x)=1}$

${\displaystyle {\text{E}}_{1}(x)=x-{\tfrac {1}{2}}}$

${\displaystyle {\text{E}}_{2}(x)=x^{2}-x=x(x-1)}$

${\displaystyle {\text{E}}_{3}(x)=x^{3}-{\tfrac {3}{2}}x^{2}+{\tfrac {1}{4}}={\tfrac {1}{4}}(2x-1)(2x^{2}-2x-1)}$

${\displaystyle {\text{E}}_{4}(x)=x^{4}-2x^{3}+x=x(x-1)(x^{2}-x-1)}$

${\displaystyle {\text{E}}_{5}(x)=x^{5}-{\tfrac {5}{2}}x^{4}+{\tfrac {5}{2}}x^{2}-{\tfrac {1}{2}}={\tfrac {1}{2}}(2x-1)(x^{2}-x-1)^{2}}$

${\displaystyle {\text{E}}_{6}(x)=x^{6}-3x^{5}+5x^{3}-3x=x(x-1)(x^{4}-2x^{3}-2x^{2}+3x+3)}$

But you can also add them to and then use the equation ${\displaystyle {\text{E}}_{0}(x)=1}$ ${\displaystyle n\in \mathbb {N} }$

${\displaystyle {\text{E}}_{n}(x)=\int _{c}^{x}n{\text{E}}_{n-1}(t)\,{\text{d}}t}$

Define inductively, where the lower limit of integration is 1/2 for odd and zero for even . ${\displaystyle c}$ ${\displaystyle n}$ ${\displaystyle n}$

The Euler's polynomials are symmetric about , i. H. ${\displaystyle {\tfrac {1}{2}}}$

${\displaystyle {\text{E}}_{n}({\tfrac {1}{2}}+x)=(-1)^{n}{\text{E}}_{n}({\tfrac {1}{2}}-x)\qquad {bzw.}\qquad {\text{E}}_{n}(x+1)=(-1)^{n}{\text{E}}_{n}(-x)}$

and their functional values ​​in the places and the relationship ${\displaystyle {\tfrac {1}{2}}}$ ${\displaystyle 0}$

${\displaystyle {\text{E}}_{n}({\tfrac {1}{2}})=2^{-n}E_{n}}$

and

${\displaystyle {\text{E}}_{n-1}(0)=(2^{n+1}-2){\frac {B_{n}}{n}}}$

suffice, whereby the Bernoulli number denotes the second kind. We also have the identity ${\displaystyle B_{n}}$

${\displaystyle {\text{E}}_{n}(x+1)+{\text{E}}_{n}(x)=2x^{n}}$

Euler's polynomial has fewer than n real zeros for n> 5. So it has five (but two doubles, i.e. only three different ones), but only the two (trivial) zeros at 0 and at 1. Let be the set of zeros . Then ${\displaystyle {\text{E}}_{n}}$ ${\displaystyle {\text{E}}_{5}}$ ${\displaystyle {\text{E}}_{6}}$ ${\displaystyle R(n)=\{x\in \mathbb {R} \colon {\text{E}}_{n}(x)=0\}}$

${\displaystyle -{\tfrac {1}{2}}|R(n)|+1\leq \min R(n)\leq \max R(n)\leq {\tfrac {1}{2}}|R(n)|}$

- where in the case n = 5 the number is to be assessed as 5, since the zeros must be counted with their multiplicity - and it applies ${\displaystyle |R(5)|}$

${\displaystyle \lim _{n\to \infty }{\frac {|R(n)|}{n}}={\frac {2}{\pi e}}\approx 0{,}2342}$

where the function actually indicates the number of elements applied to a set. ${\displaystyle |\cdot |}$

Occurrence

Taylor series

The sequence of Euler's numbers occurs, for example, in the Taylor expansion of ${\displaystyle E_{n}}$

${\displaystyle \sec(x)={\frac {1}{\cos(x)}}={\frac {1}{\cosh(\mathrm {i} \,x)}}=\sum _{n=0}^{\infty }(-1)^{n}E_{n}{\frac {x^{n}}{n!}}}$

on. It is related to the sequence of Bernoulli numbers, which is what you can see in the representation ${\displaystyle B_{n}}$

${\displaystyle \operatorname {csch} (x)={\frac {1}{\sinh(x)}}=\sum _{n=0}^{\infty }(2-2^{n})B_{n}{\frac {x^{n-1}}{n!}}}$

recognizes. From the radius of convergence of the Taylor expansion of the secant function - the cosine in the denominator becomes 0 at - from follows from the root criterion that must apply asymptotically. Of course, they also appear in the Taylor series of the higher derivatives of the hyperbolic secant or the Gudermann function . ${\displaystyle {\tfrac {\pi }{2}}}$ ${\displaystyle {\tfrac {\pi }{2}}}$ ${\displaystyle \limsup \log |{\tfrac {E_{n}}{n!}}|\sim n\log \left({\tfrac {2}{\pi }}\right)}$

Integrals

They also appear with some improper integrals ; for example with the integral

${\displaystyle \int \limits _{0}^{\infty }{\frac {\ln ^{n}(x)}{1+x^{2}}}\,dx=|E_{n}|\left({\frac {\pi }{2}}\right)^{n+1}}$.

Permutations

Euler's numbers appear when counting the number of alternating permutations with an even number of elements. An alternate permutation of values a list of these values , so that these permutation no triples with containing the sorted is. In general, the number of alternating permutations of elements (which are comparable) ${\displaystyle a_{1},a_{2},\ldots ,a_{2n}}$ ${\displaystyle a_{j-1},a_{j},a_{j+1}}$ ${\displaystyle 1<j<2n}$ ${\displaystyle A_{2n}}$ ${\displaystyle 2n}$

${\displaystyle A_{2n}=2|E_{2n}|}$,

where the factor of two arises from the fact that each permutation can be converted into another alternating permutation by reversing the order. The following applies to any (i.e. also odd) number ${\displaystyle n\in \mathbb {N} _{0}}$

${\displaystyle A_{n}=2\,n!\,\alpha _{n}}$

with and ${\displaystyle \alpha _{0}=\alpha _{1}=1}$

${\displaystyle \alpha _{n}={\frac {1}{2n}}\sum _{j=0}^{n-1}\alpha _{j}\alpha _{n-1-j}}$

for , which gives another efficient algorithm for determining the . For odd numbers, the values ​​are also called tangent numbers. ${\displaystyle n\geq 2}$ ${\displaystyle E_{2n}}$ ${\displaystyle n}$ ${\displaystyle {\tfrac {A_{n}}{2}}}$

literature

• JM Borwein, PB Borwein, K. Dilcher, Pi, Euler Numbers, and Asymptotic Expansions , AMM, V. 96, no. 8, (Oct. 1989), pp. 681-687

Individual evidence

1. ↑ M. Abramowitz and I. Stegun, Handbook of Mathematical Functions . Dover, NY 1964, p. 807

Web links

• NIST Digital Library of Mathematical Functions
• Eric W. Weisstein : Euler's Numbers . In: MathWorld (English).
• Follow A122045 in OEIS