DLMF: Untitled Document

… ►

Bernoulli Numbers and Polynomials

►The origin of the notation

B_{n}

,

B_{n}\left(x\right)

, is not clear. …

§24.14 Sums

… ►

24.14.2

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

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Permalink:: http://dlmf.nist.gov/24.14.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(i), §24.14 and Ch.24

… ►

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}.

ⓘ

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

… ►

24.14.8

\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!}B_{2j}B_{2k}B_{2\ell}=(n-1)(2n-1)B_{2n}+n(% n-\tfrac{1}{2})B_{2n-2},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli 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.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

… ►

24.14.10

\sum\frac{(2n)!}{(2j)!(2k)!(2\ell)!(2m)!}B_{2j}B_{2k}B_{2\ell}B_{2m}=-{2n+3% \choose 3}B_{2n}-\frac{4}{3}n^{2}(2n-1)B_{2n-2}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $j$ : integer, $k$ : integer, $\ell$ : integer, $m$ : integer and $n$ : integer
Referenced by:: §24.14(ii)
Permalink:: http://dlmf.nist.gov/24.14.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §24.14(ii), §24.14 and Ch.24

…

… ►Equations (24.5.3) and (24.5.4) enable

B_{n}

and

E_{n}

to be computed by recurrence. … ►

B_{2n}=\dfrac{N_{2n}}{D_{2n}}.

… ►For algorithms for computing

B_{n}

,

E_{n}

,

B_{n}\left(x\right)

, and

E_{n}\left(x\right)

see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). ►

§24.19(ii) Values of $B_{n}$ Modulo $p$

►For number-theoretic applications it is important to compute

B_{2n}\pmod{p}

for

2n\leq p-3

; in particular to find the irregular pairs

(2n,p)

for which

B_{2n}\equiv 0\pmod{p}

. …

… ►

24.10.1

B_{2n}+\sum_{(p-1)\mathbin{|}2n}\frac{1}{p}=\hbox{integer},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $n$ : integer and $p$ : prime
Permalink:: http://dlmf.nist.gov/24.10.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(i), §24.10 and Ch.24

… ►The denominator of

B_{2n}

is the product of all these primes

p

. ►

24.10.2

pB_{2n}\equiv p-1\pmod{p^{\ell+1}},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\equiv$ : modular equivalence, $\ell$ : integer, $n$ : integer and $p$ : prime
Referenced by:: §24.10(i)
Permalink:: http://dlmf.nist.gov/24.10.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(i), §24.10 and Ch.24

… ►

24.10.3

\frac{B_{m}}{m}\equiv\frac{B_{n}}{n}\pmod{p},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\equiv$ : modular equivalence, $m$ : integer, $n$ : integer and $p$ : prime
Permalink:: http://dlmf.nist.gov/24.10.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(ii), §24.10 and Ch.24

… ►

24.10.4

(1-p^{m-1})\frac{B_{m}}{m}\equiv(1-p^{n-1})\frac{B_{n}}{n}\pmod{p^{\ell+1}},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\equiv$ : modular equivalence, $\ell$ : integer, $m$ : integer, $n$ : integer and $p$ : prime
Permalink:: http://dlmf.nist.gov/24.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.10(ii), §24.10 and Ch.24

…

… ►

§24.5(ii) Other Identities

►

24.5.6

\sum_{k=2}^{n}{n\choose k-2}\frac{B_{k}}{k}=\frac{1}{(n+1)(n+2)}-B_{n+1},

n=2,3,\dots

,

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Referenced by:: §24.5(ii)
Permalink:: http://dlmf.nist.gov/24.5.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §24.5(ii), §24.5 and Ch.24

►

24.5.7

\sum_{k=0}^{n}{n\choose k}\frac{B_{k}}{n+2-k}=\frac{B_{n+1}}{n+1},

n=1,2,\dots

,

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer
Referenced by:: §24.5(ii)
Permalink:: http://dlmf.nist.gov/24.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §24.5(ii), §24.5 and Ch.24

►

24.5.8

\sum_{k=0}^{n}\frac{2^{2k}B_{2k}}{(2k)!(2n+1-2k)!}=\frac{1}{(2n)!},

n=1,2,\dots

.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $!$ : factorial (as in $n!$ ), $k$ : integer and $n$ : integer
Referenced by:: §24.5(ii)
Permalink:: http://dlmf.nist.gov/24.5.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §24.5(ii), §24.5 and Ch.24

►

§24.5(iii) Inversion Formulas

…

… ►In (4.19.3)–(4.19.9),

B_{n}

are the Bernoulli numbers and

E_{n}

are the Euler numbers (§§24.2(i)–24.2(ii)). ►

4.19.3

\tan z=z+\frac{z^{3}}{3}+\frac{2}{15}z^{5}+\frac{17}{315}z^{7}+\cdots+\frac{(-% 1)^{n-1}2^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,

|z|<\frac{1}{2}\pi

,

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\tan\NVar{z}$ : tangent function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.67
Referenced by:: §4.19
Permalink:: http://dlmf.nist.gov/4.19.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

►

4.19.4

\csc z=\frac{1}{z}+\frac{z}{6}+\frac{7}{360}z^{3}+\frac{31}{15120}z^{5}+\cdots% +\frac{(-1)^{n-1}2(2^{2n-1}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,

0<|z|<\pi

,

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$ : cosecant function, $!$ : factorial (as in $n!$ ), $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.68
Referenced by:: §4.33
Permalink:: http://dlmf.nist.gov/4.19.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

… ►

4.19.6

\cot z=\frac{1}{z}-\frac{z}{3}-\frac{z^{3}}{45}-\frac{2}{945}z^{5}-\cdots-% \frac{(-1)^{n-1}2^{2n}B_{2n}}{(2n)!}z^{2n-1}-\cdots,

0<|z|<\pi

,

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cot\NVar{z}$ : cotangent function, $!$ : factorial (as in $n!$ ), $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.70
Permalink:: http://dlmf.nist.gov/4.19.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

►

4.19.7

\ln\left(\frac{\sin z}{z}\right)=\sum_{n=1}^{\infty}\frac{(-1)^{n}2^{2n-1}B_{2% n}}{n(2n)!}z^{2n},

|z|<\pi

,

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.3.71
Permalink:: http://dlmf.nist.gov/4.19.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §4.19 and Ch.4

…

… ►

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

…

… ►

§24.15(i) Genocchi Numbers

… ►

24.15.2

G_{n}=2(1-2^{n})B_{n}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $n$ : integer and $G_{n}$ : Genocchi numbers
Permalink:: http://dlmf.nist.gov/24.15.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.15(i), §24.15 and Ch.24

… ►

§24.15(ii) Tangent Numbers

… ►

§24.15(iii) Stirling Numbers

… ►

24.15.6

B_{n}=\sum_{k=0}^{n}(-1)^{k}\frac{k!S\left(n,k\right)}{k+1},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $!$ : factorial (as in $n!$ ), $k$ : integer and $n$ : integer
Referenced by:: §24.15(iii)
Permalink:: http://dlmf.nist.gov/24.15.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §24.15(iii), §24.15 and Ch.24

…

… ►

4.33.3

\tanh z=z-\frac{z^{3}}{3}+\frac{2}{15}z^{5}-\frac{17}{315}z^{7}+\cdots+\frac{2% ^{2n}(2^{2n}-1)B_{2n}}{(2n)!}z^{2n-1}+\cdots,

|z|<\frac{1}{2}\pi

.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\tanh\NVar{z}$ : hyperbolic tangent function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.5.64
Permalink:: http://dlmf.nist.gov/4.33.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.33 and Ch.4

►For

B_{2n}

see §24.2(i). …

§24.6 Explicit Formulas

… ►

24.6.1

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

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli 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.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

►

24.6.2

B_{n}=\frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{n}{\genfrac{(}{)}{0.% 0pt}{}{n+1}{k-j}}\Bigg{/}{\genfrac{(}{)}{0.0pt}{}{n}{k}},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli 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.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

… ►

24.6.9

B_{n}=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j}j^{n},

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli 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.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

… ►

24.6.11

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

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli 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.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §24.6 and Ch.24

…

Bernoulli numbers

1—10 of 59 matching pages

1: 24.1 Special Notation

Bernoulli Numbers and Polynomials

2: 24.14 Sums

§24.14 Sums

3: 24.19 Methods of Computation

§24.19(ii) Values of $B_{n}$ Modulo $p$

4: 24.10 Arithmetic Properties

5: 24.5 Recurrence Relations

§24.5(ii) Other Identities

§24.5(iii) Inversion Formulas

6: 4.19 Maclaurin Series and Laurent Series

7: 24.21 Software

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

8: 24.15 Related Sequences of Numbers

§24.15(i) Genocchi Numbers

§24.15(ii) Tangent Numbers

§24.15(iii) Stirling Numbers

9: 4.33 Maclaurin Series and Laurent Series

10: 24.6 Explicit Formulas

§24.6 Explicit Formulas

Bernoulli numbers

1—10 of 59 matching pages

1: 24.1 Special Notation

Bernoulli Numbers and Polynomials

2: 24.14 Sums

§24.14 Sums

3: 24.19 Methods of Computation

§24.19(ii) Values of B n Modulo p

4: 24.10 Arithmetic Properties

5: 24.5 Recurrence Relations

§24.5(ii) Other Identities

§24.5(iii) Inversion Formulas

6: 4.19 Maclaurin Series and Laurent Series

7: 24.21 Software

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

8: 24.15 Related Sequences of Numbers

§24.15(i) Genocchi Numbers

§24.15(ii) Tangent Numbers

§24.15(iii) Stirling Numbers

9: 4.33 Maclaurin Series and Laurent Series

10: 24.6 Explicit Formulas

§24.6 Explicit Formulas

§24.19(ii) Values of $B_{n}$ Modulo $p$

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