The Euler numbers or sometimes Euler numbers (by Leonhard Euler are) a sequence of integers, by the Taylor expansion of the hyperbolic secant hyperbolic${\ displaystyle \, E_ {n}}$
 ${\ displaystyle \ operatorname {six} (x) = {\ frac {1} {\ cosh (x)}} = {\ frac {2} {e ^ {x} + e ^ { x}}} = \ sum _ {n = 0} ^ {\ infty} E_ {n} {\ frac {x ^ {n}} {n!}}}$
are defined. They are not to be confused with the twoparameter 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 easytoremember form of the recursion equation with the seed is
${\ displaystyle E_ {0} = 1}$
 ${\ displaystyle \ forall \, n \ in \ mathbb {N} \ colon \ quad (E + 1) ^ {n} + (E1) ^ {n} = 0}$
where is to be interpreted as and from what
${\ displaystyle E ^ {n}}$${\ displaystyle E_ {n}}$
 ${\ displaystyle \ forall \, n \ in \ mathbb {N} \ colon \ quad \ sum _ {k = 0} ^ {n} \ left (1 + ( 1) ^ {nk} \ right) {n \ choose k} E_ {k} = 0}$
or the explicit form through index transformation
 ${\ displaystyle \ forall \, n \ in \ mathbb {N} \ colon \ quad E_ {n} =  \ sum _ {k = 1} ^ {\ lfloor n / 2 \ rfloor} {n \ choose 2k} E_ {n2k}}$
follows.
Closed representations
The Euler's numbers can even be made exact
 ${\ displaystyle \ forall \, n \ in \ mathbb {N} _ {0} \ colon \ quad E_ {2n} = {\ frac {(2n)! \, 2 ^ {2n + 2}} {( 1 ) ^ {n} \ pi ^ {2n + 1}}} \ sum _ {k = 0} ^ {\ infty} {\ frac {(1) ^ {k}} {(2k + 1) ^ {2n +1}}} = {\ frac {(2n)! \, 2} {( 1) ^ {n} (2 \ pi) ^ {2n + 1}}} \ left (\ zeta (2n + 1, {\ tfrac {1} {4}})  \ zeta (2n + 1, {\ tfrac {3} {4}}) \ right)}$
using the Hurwitz zeta function if is, represent. And using its functional equation (there with m = 1, n = 4) the elegant relationship
${\ displaystyle \ zeta}$${\ 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
 ${\ displaystyle {\ text {E}} _ {0} (x) = 1}$
 ${\ displaystyle {\ text {E}} _ {1} (x) = x  {\ tfrac {1} {2}}}$
 ${\ displaystyle {\ text {E}} _ {2} (x) = x ^ {2} x = x (x1)}$
 ${\ displaystyle {\ text {E}} _ {3} (x) = x ^ {3}  {\ tfrac {3} {2}} x ^ {2} + {\ tfrac {1} {4}} = {\ tfrac {1} {4}} (2x1) (2x ^ {2} 2x1)}$
 ${\ displaystyle {\ text {E}} _ {4} (x) = x ^ {4} 2x ^ {3} + x = x (x1) (x ^ {2} x1)}$
 ${\ displaystyle {\ text {E}} _ {5} (x) = x ^ {5}  {\ tfrac {5} {2}} x ^ {4} + {\ tfrac {5} {2}} x ^ {2}  {\ tfrac {1} {2}} = {\ tfrac {1} {2}} (2x1) (x ^ {2} x1) ^ {2}}$
 ${\ displaystyle {\ text {E}} _ {6} (x) = x ^ {6} 3x ^ {5} + 5x ^ {3} 3x = x (x1) (x ^ {4} 2x ^ {3} 2x ^ {2} + 3x + 3)}$
But you can also add them to and then use the equation
${\ displaystyle {\ text {E}} _ {0} (x) = 1}$${\ displaystyle n \ in \ mathbb {N}}$
 ${\ displaystyle {\ text {E}} _ {n} (x) = \ int _ {c} ^ {x} n {\ text {E}} _ {n1} (t) \, {\ text {d}} t}$
Define inductively, where the lower limit of integration is 1/2 for odd and zero for even .
${\ displaystyle c}$${\ displaystyle n}$${\ displaystyle n}$
The Euler's polynomials are symmetric about , i. H.
${\ displaystyle {\ tfrac {1} {2}}}$
 ${\ displaystyle {\ text {E}} _ {n} ({\ tfrac {1} {2}} + x) = ( 1) ^ {n} {\ text {E}} _ {n} ({ \ tfrac {1} {2}}  x) \ qquad {or} \ qquad {\ text {E}} _ {n} (x + 1) = ( 1) ^ {n} {\ text {E }} _ {n} ( x)}$
and their functional values in the places and the relationship
${\ displaystyle {\ tfrac {1} {2}}}$${\ displaystyle 0}$
 ${\ displaystyle {\ text {E}} _ {n} ({\ tfrac {1} {2}}) = 2 ^ { n} E_ {n}}$
and
 ${\ displaystyle {\ text {E}} _ {n1} (0) = (2 ^ {n + 1} 2) {\ frac {B_ {n}} {n}}}$
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}}$${\ displaystyle R (n) = \ {x \ in \ mathbb {R} \ colon {\ text {E}} _ {n} (x) = 0 \}}$
 ${\ displaystyle  {\ tfrac {1} {2}}  R (n)  +1 \ leq \ min R (n) \ leq \ max R (n) \ leq {\ tfrac {1} {2}}  R (n) }$
 where in the case n = 5 the number is to be assessed as 5, since the zeros must be counted with their multiplicity  and it applies
${\ displaystyle  R (5) }$
 ${\ displaystyle \ lim _ {n \ to \ infty} {\ frac { R (n) } {n}} = {\ frac {2} {\ pi e}} \ approx 0 {,} 2342}$
where the function actually indicates the number of elements applied to a set.
${\ displaystyle  \ cdot }$
Occurrence
Taylor series
The sequence of Euler's numbers occurs, for example, in the Taylor expansion of
${\ displaystyle E_ {n}}$
 ${\ displaystyle \ sec (x) = {\ frac {1} {\ cos (x)}} = {\ frac {1} {\ cosh (\ mathrm {i} \, x)}} = \ sum _ { n = 0} ^ {\ infty} ( 1) ^ {n} E_ {n} {\ frac {x ^ {n}} {n!}}}$
on. It is related to the sequence of Bernoulli numbers, which is what you can see in the representation
${\ displaystyle B_ {n}}$
 ${\ displaystyle \ operatorname {csch} (x) = {\ frac {1} {\ sinh (x)}} = \ sum _ {n = 0} ^ {\ infty} (22 ^ {n}) B_ {n} {\ frac {x ^ {n1}} {n!}}}$
recognizes. From the radius of convergence of the Taylor expansion of the secant function  the cosine in the denominator becomes 0 at  from follows from the root criterion that must apply asymptotically. Of course, they also appear in the Taylor series of the higher derivatives of the hyperbolic secant or the Gudermann function .
${\ displaystyle {\ tfrac {\ pi} {2}}}$${\ 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}}$${\ displaystyle a_ {j1}, a_ {j}, a_ {j + 1}}$${\ 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
${\ displaystyle \ alpha _ {0} = \ alpha _ {1} = 1}$
 ${\ displaystyle \ alpha _ {n} = {\ frac {1} {2n}} \ sum _ {j = 0} ^ {n1} \ alpha _ {j} \ alpha _ {n1j} }$
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. 681687
Individual evidence

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