DLMF: Untitled Document

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Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations

E_{n}

,

E_{n}\left(x\right)

, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

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§24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , and $E_{n}\left(x\right)$

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§24.6 Explicit Formulas

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24.6.4

E_{2n}=\sum_{k=1}^{n}\frac{1}{2^{k-1}}\sum_{j=1}^{k}(-1)^{j}{2k\choose k-j}j^{% 2n},

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

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24.6.5

E_{2n}=\frac{1}{2^{n-1}}\sum_{k=0}^{n-1}(-1)^{n-k}(n-k)^{2n}\*\sum_{j=0}^{k}{2% n-2j\choose k-j}2^{j},

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

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24.6.10

E_{n}=\frac{1}{2^{n}}\sum_{k=1}^{n+1}{n+1\choose k}\sum_{j=0}^{k-1}(-1)^{j}(2j% +1)^{n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6, §24.6
Permalink:: http://dlmf.nist.gov/24.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

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24.6.12

E_{2n}=\sum_{k=0}^{2n}\frac{1}{2^{k}}\sum_{j=0}^{k}(-1)^{j}{k\choose j}(1+2j)^% {2n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6, §24.6
Permalink:: http://dlmf.nist.gov/24.6.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

§24.14 Sums

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24.14.4

\sum_{k=0}^{n}{n\choose k}E_{k}E_{n-k}=-2^{n+1}E_{n+1}\left(0\right)=-2^{n+2}(% 1-2^{n+2})\frac{B_{n+2}}{n+2}.

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(i), §24.14 and Ch.24

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24.14.6

\sum_{k=0}^{n}{n\choose k}2^{k}B_{k}E_{n-k}=2(1-2^{n-1})B_{n}-nE_{n-1}.

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Referenced by:: §24.14(i)
Permalink:: http://dlmf.nist.gov/24.14.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(i), §24.14 and Ch.24

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24.14.9

\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!}E_{2j}E_{2k}E_{2\ell}=\tfrac{1}{2}\left(E_% {2n}-E_{2n+2}\right).

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $!$ : factorial (as in $n!$ ), $j$ : integer, $k$ : integer, $\ell$ : integer and $n$ : integer
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

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24.14.12

\det[E_{r+s}]=(-1)^{n(n+1)/2}\left(\prod_{k=1}^{n}k!\right)^{2}.

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\det$ : determinant, $!$ : factorial (as in $n!$ ), $k$ : integer, $n$ : integer, $r$ : row and $s$ : column
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

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§24.10 Arithmetic Properties

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§24.10(ii) Kummer Congruences

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24.10.5

E_{n}\equiv E_{n+p-1}\pmod{p},

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\equiv$ : modular equivalence, $n$ : integer and $p$ : prime
Referenced by:: §24.10(ii)
Permalink:: http://dlmf.nist.gov/24.10.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(ii), §24.10 and Ch.24

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24.10.6

E_{2n}\equiv E_{2n+w}\pmod{2^{\ell}},

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\equiv$ : modular equivalence, $\ell$ : integer and $n$ : integer
Referenced by:: §24.10(ii)
Permalink:: http://dlmf.nist.gov/24.10.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(ii), §24.10 and Ch.24

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§24.10(iv) Factors

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§24.5 Recurrence Relations

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24.5.4

\sum_{k=0}^{n}{2n\choose 2k}E_{2k}=0,

n=1,2,\dots

,

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Referenced by:: §24.19(i)
Permalink:: http://dlmf.nist.gov/24.5.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.5(i), §24.5 and Ch.24

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24.5.5

\sum_{k=0}^{n}{n\choose k}2^{k}E_{n-k}+E_{n}=2.

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.5.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §24.5(i), §24.5 and Ch.24

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§24.5(iii) Inversion Formulas

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b_{n}=\sum_{k=0}^{\left\lfloor\ifrac{n}{2}\right\rfloor}{n\choose 2k}E_{2k}a_{% n-2k}.

§24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of

N_{n}

and

E_{n}

for

n=20(2)60

and

n=8(2)42

, respectively. …

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§24.2(ii) Euler Numbers and Polynomials

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E_{2n+1}=0

,

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Table 24.2.1: Bernoulli and Euler numbers.

► ►►

$n$	$B_{n}$	$E_{n}$
…

► … ►

Table 24.2.4: Euler numbers

E_{n}

.

► ►►

$n$	$E_{n}$
…

► …

§24.9 Inequalities

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24.9.3

4^{-n}|E_{2n}|>(-1)^{n}E_{2n}\left(x\right)>0,

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex
A&S Ref:: 23.1.13
Referenced by:: §24.9, §24.9
Permalink:: http://dlmf.nist.gov/24.9.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.9 and Ch.24

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24.9.7

8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}\left(1+\frac{1}{12n}% \right)>(-1)^{n}E_{2n}>8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $n$ : integer
Referenced by:: §24.9, §24.9
Permalink:: http://dlmf.nist.gov/24.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §24.9 and Ch.24

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24.9.10

\frac{4^{n+1}(2n)!}{\pi^{2n+1}}>(-1)^{n}E_{2n}>\frac{4^{n+1}(2n)!}{\pi^{2n+1}}% \frac{1}{1+3^{-1-2n}}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ) and $n$ : integer
A&S Ref:: 23.1.15
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.9.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §24.9 and Ch.24

§24.11 Asymptotic Approximations

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24.11.3

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

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Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ) and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

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24.11.4

(-1)^{n}E_{2n}\sim 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm and $n$ : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.11 and Ch.24

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Euler numbers

1—10 of 100 matching pages

1: 24.1 Special Notation

Euler Numbers and Polynomials

2: 24.21 Software

§24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , and $E_{n}\left(x\right)$

3: 24.6 Explicit Formulas

§24.6 Explicit Formulas

4: 24.14 Sums

§24.14 Sums

5: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

§24.10(ii) Kummer Congruences

§24.10(iv) Factors

6: 24.5 Recurrence Relations

§24.5 Recurrence Relations

§24.5(iii) Inversion Formulas

7: 24.20 Tables

§24.20 Tables

8: 24.2 Definitions and Generating Functions

§24.2(ii) Euler Numbers and Polynomials

9: 24.9 Inequalities

§24.9 Inequalities

10: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

Euler numbers

1—10 of 100 matching pages

1: 24.1 Special Notation

Euler Numbers and Polynomials

2: 24.21 Software

§24.21(ii) B n , B n ⁡ ( x ) , E n , and E n ⁡ ( x )

3: 24.6 Explicit Formulas

§24.6 Explicit Formulas

4: 24.14 Sums

§24.14 Sums

5: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

§24.10(ii) Kummer Congruences

§24.10(iv) Factors

6: 24.5 Recurrence Relations

§24.5 Recurrence Relations

§24.5(iii) Inversion Formulas

7: 24.20 Tables

§24.20 Tables

8: 24.2 Definitions and Generating Functions

§24.2(ii) Euler Numbers and Polynomials

9: 24.9 Inequalities

§24.9 Inequalities

10: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

§24.21(ii) $B_{n}$ , $B_{n}\left(x\right)$ , $E_{n}$ , and $E_{n}\left(x\right)$