Euler number

The Euler numbers or sometimes Euler numbers (after Leonhard Euler ) are a sequence of integers, the hyperbolic by the Taylor expansion of the hyperbolic secant

Are defined. The wording is not to be confused with the ( unique) Euler's number or even the two-parameter Euler numbers E (n, k) that are combinatorially with which they are associated.

• 4.1 Taylor series
• 4.2 Integral
• 4.3 permutations

Numerical values

Are the first Euler numbers ≠ 0

All Euler numbers with odd index are zero, while those with even index alternating in sign. Also have the positive values ​​, with the exception of E0 division by 10 the rest 5, and the negative values ​​modulo 10 value of -1 and the rest 9

Some authors have the numbers with odd index completely gone, halve the indexes, so to speak, since there the values ​​are not considered to 0, and define their Euler numbers as the remaining sequence. Sometimes the Euler numbers are also defined so that they are all positive, ie correspond to ours.

Properties

Asymptotic behavior

For the asymptotic behavior of the Euler numbers

Or more precisely

With the ~ - equivalence notation.

Recurrence

An easy-to -remember form of the recurrence equation with the initial value is

Where as is to be interpreted and what

Or through index transformation, the explicit form

Followed.

Closed representations

The Eulerian numbers can even be exactly

By means of the Hurwitz zeta function if is representing. And by utilizing their functional equation (where m = 1, n = 4), the relationship elegant

Establish that identifies these numbers as scaled values ​​of this function on holomorphic function. Thus we obtain

Which establishes a direct connection with the Bernoulli polynomials and thus the Bernoulli numbers.

Euler polynomials

The Euler polynomials are generating most of their function

Implicitly defined. The first loud

But one can and then for about the equation also

Inductively define the lower limit of integration is odd 1/2, and is zero for straight.

The Euler polynomials are symmetric about, ie

And their function values ​​at the points and the relationship

And

Suffice where Bn denotes the Bernoulli number of the second type. Further, we have the identity

The Eulerian polynomial has for n > 5 less than n real zeros. Thus, although has five ( two double, however, say only three different ones), but even just the two ( trivial ) zeros at 0 and at 1 Be the zero set. Then

- And in the case n = 5, the number 5 is to evaluate, since the zeros have to be counted with multiplicities - and it is

Where the function applied to a lot actually indicates the element number.

Occurrence

Taylor series

The sequence of the Euler numbers, for example, occurs in the Taylor development of

On. It is related to the sequence of Bernoulli numbers what is in the graphical

Recognizes. From the radius of convergence of the Taylor expansion of the secant function - the cosine in the denominator there is at 0 - and must apply by asymptotically follows from the root test. They occur naturally in the Taylor series of the higher derivatives of the hyperbolic secant or Gudermann function.

Integrals

Also in some improper integrals they occur; For example, when the integral

Permutations

The Euler numbers occur with even number of elements in counting the number of alternating permutations. An alternating permutation of values ​​is a list of these values ​​, so that this permutation contains no triple with that is ordered. The following applies for the number of permutations of alternating elements ( comparable )

Where the factor of two arises because one can lead to another alternating permutation each permutation by reversing the order. For an arbitrary (also odd) number

With and

For, which is also another efficient algorithm for determining who receives. For odd values ​​are also called tangent numbers.