Bernoulli numbers

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11: 24.4 Basic Properties

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24.4.7 $$\sum _{k=1}^m{k}^n=\frac{B_{n+1}\left(m+1\right)-B_{n+1}}{n+1},$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $k$: integer, $m$: integer and $n$: integer

A&S Ref:

23.1.4

Referenced by:

§24.17(iii)

Permalink:

http://dlmf.nist.gov/24.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

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§24.4(iv) Finite Expansions

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24.4.25 $$B_n\left(0\right)={(-1)}^n{B}_n\left(1\right)=B_n.$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials and $n$: integer

A&S Ref:

23.1.20

Referenced by:

§24.17(ii), §24.4(vi)

Permalink:

http://dlmf.nist.gov/24.4.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

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24.4.27 $$B_n\left(\frac{1}{2}\right)=-(1-2^{1-n})B_n,$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials and $n$: integer

A&S Ref:

23.1.21

Referenced by:

§2.10(i), §24.17(ii)

Permalink:

http://dlmf.nist.gov/24.4.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

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§24.4(ix) Relations to Other Functions

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12: 24.9 Inequalities

§24.9 Inequalities

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24.9.1 $$|B_{2n}|>|B_{2n}\left(x\right)|,$$ $1>x>0$,

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $n$: integer and $x$: real or complex

A&S Ref:

23.1.13

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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24.9.2 $$(2-2^{1-2n})|B_{2n}|\ge |B_{2n}\left(x\right)-B_{2n}|,$$ $1\ge x\ge 0$.

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $n$: integer and $x$: real or complex

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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24.9.6 $$5\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n}>{(-1)}^{n+1}{B}_{2n}>4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

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Symbols:

$B_n$: Bernoulli 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.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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24.9.8 $$\frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{\beta -2n}}\ge {(-1)}^{n+1}{B}_{2n}\ge \frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{-2n}}$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $\beta$: quantity

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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13: 24.7 Integral Representations

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§24.7(i) Bernoulli and Euler Numbers

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24.7.2 $$B_{2n}={(-1)}^{n+1}4n{\int }_0^{\mathrm{\infty }}\frac{t^{2n-1}}{{\mathrm{e}}^{2\pi t}-1}dt={(-1)}^{n+1}2n{\int }_0^{\mathrm{\infty }}{t}^{2n-1}{\mathrm{e}}^{-\pi t}\mathrm{csch}\left(\pi t\right)dt,$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(i), §24.7 and Ch.24

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24.7.3 $$B_{2n}={(-1)}^{n+1}\frac{\pi }{1-2^{1-2n}}{\int }_0^{\mathrm{\infty }}{t}^{2n}{\mathrm{sech}}^2\left(\pi t\right)dt,$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{sech}z$: hyperbolic secant function, $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(i), §24.7 and Ch.24

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24.7.4 $$B_{2n}={(-1)}^{n+1}\pi {\int }_0^{\mathrm{\infty }}{t}^{2n}{\mathrm{csch}}^2\left(\pi t\right)dt,$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{csch}z$: hyperbolic cosecant function, $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(i), §24.7 and Ch.24

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24.7.5 $$B_{2n}={(-1)}^n\frac{2n(2n-1)}{\pi }{\int }_0^{\mathrm{\infty }}{t}^{2n-2}\ln \left(1-{\mathrm{e}}^{-2\pi t}\right)dt.$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function, $n$: integer and $t$: real or complex

Referenced by:

§24.7(i)

Permalink:

http://dlmf.nist.gov/24.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(i), §24.7 and Ch.24

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14: 24.13 Integrals

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24.13.4 $${\int }_0^{1/2}{B}_n\left(t\right)dt=\frac{1-2^{n+1}}{2^n}\frac{B_{n+1}}{n+1},$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(i), §24.13 and Ch.24

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24.13.6 $${\int }_0^1{B}_n\left(t\right)B_m\left(t\right)dt=\frac{{(-1)}^{n-1}m!n!}{(m+n)!}{B}_{m+n}.$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $m$: integer, $n$: integer and $t$: real or complex

A&S Ref:

23.1.12

Referenced by:

§24.13(i)

Permalink:

http://dlmf.nist.gov/24.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(i), §24.13 and Ch.24

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24.13.8 $${\int }_0^1{E}_n\left(t\right)dt=-2\frac{E_{n+1}\left(0\right)}{n+1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}{B}_{n+2},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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24.13.9 $${\int }_0^{1/2}{E}_{2n}\left(t\right)dt=-\frac{E_{2n+1}\left(0\right)}{2n+1}=\frac{2(2^{2n+2}-1)B_{2n+2}}{(2n+1)(2n+2)},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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24.13.11 $${\int }_0^1{E}_n\left(t\right)E_m\left(t\right)dt={(-1)}^{n}4\frac{(2^{m+n+2}-1)m!n!}{(m+n+2)!}{B}_{m+n+2}.$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $m$: integer, $n$: integer and $t$: real or complex

A&S Ref:

23.1.12

Referenced by:

§24.13(ii)

Permalink:

http://dlmf.nist.gov/24.13.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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15: 24.2 Definitions and Generating Functions

§24.2 Definitions and Generating Functions

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$B_{2n+1}=0$ ,

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Table 24.2.1: Bernoulli and Euler numbers.

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$n$	$B_n$	$E_n$
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Table 24.2.3: Bernoulli numbers $B_n=N/D$.

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$n$	$N$	$D$	$B_n$
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16: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

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24.11.1 $${(-1)}^{n+1}{B}_{2n}\sim \frac{2(2n)!}{{(2\pi )}^{2n}},$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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24.11.2 $${(-1)}^{n+1}{B}_{2n}\sim 4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

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Symbols:

$B_n$: Bernoulli 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

Permalink:

http://dlmf.nist.gov/24.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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24.11.4 $${(-1)}^n{E}_{2n}\sim 8\sqrt{\frac{n}{\pi }}{\left(\frac{4n}{\pi \mathrm{e}}\right)}^{2n}.$$

ⓘ

Symbols:

$E_n$: 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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17: 25.11 Hurwitz Zeta Function

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25.11.22 $${\zeta }^{\prime }(1-2n,\frac{1}{2})=-\frac{B_{2n}\ln 2}{n\cdot 4^n}-\frac{(2^{2n-1}-1){\zeta }^{\prime }\left(1-2n\right)}{2^{2n-1}},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Keywords:

derivative

Source:

Miller and Adamchik (1998, (17), p. 205)

Permalink:

http://dlmf.nist.gov/25.11.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

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25.11.34 $$n{\int }_0^a{\zeta }^{\prime }(1-n,x)dx={\zeta }^{\prime }(-n,a)-{\zeta }^{\prime }\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a\right)}{n(n+1)},$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>0$.

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $a$: real or complex parameter

Keywords:

definite integral, integral representation

Source:

Adamchik (1998, (15), p. 196)

Permalink:

http://dlmf.nist.gov/25.11.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(viii), §25.11 and Ch.25

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25.11.43 $$\zeta (s,a)-\frac{a^{1-s}}{s-1}-\frac{1}{2}{a}^{-s}\sim \sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}.$$

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Symbols:

$B_n$: Bernoulli numbers, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $k$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable

Keywords:

asymptotic approximation

Source:

Paris (2005b, (1.3), (1.4), p. 298)

Referenced by:

§25.11(xii), §25.11(xii), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.11.E43

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{(2k)!}\frac{\mathrm{\Gamma }\left(s+2k-1\right)}{\mathrm{\Gamma }\left(s\right)}{a}^{1-s-2k}$ more concisely as ${\sum }_{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}$ using the Pochhammer symbol.

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

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25.11.44 $${\zeta }^{\prime }(-1,a)-\frac{1}{12}+\frac{1}{4}{a}^2-\left(\frac{1}{12}-\frac{1}{2}a+\frac{1}{2}{a}^2\right)\ln a\sim -\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{(2k+2)(2k+1)2k}{a}^{-2k},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\sim$: Poincaré asymptotic expansion, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $a$: real or complex parameter

Keywords:

asymptotic approximation

Source:

Elizalde (1986, (18), p. 349)

Referenced by:

§25.11(xii)

Permalink:

http://dlmf.nist.gov/25.11.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

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25.11.45 $${\zeta }^{\prime }(-2,a)-\frac{1}{12}a+\frac{1}{9}{a}^3-\left(\frac{1}{6}a-\frac{1}{2}{a}^2+\frac{1}{3}{a}^3\right)\ln a\sim \sum _{k=1}^{\mathrm{\infty }}\frac{2B_{2k+2}}{(2k+2)(2k+1)2k(2k-1)}{a}^{-(2k-1)}.$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\sim$: Poincaré asymptotic expansion, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $a$: real or complex parameter

Keywords:

asymptotic approximation

Source:

Elizalde (1986, (19), p. 350)

Referenced by:

§25.11(xii)

Permalink:

http://dlmf.nist.gov/25.11.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

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18: 25.1 Special Notation

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$k,m,n$	nonnegative integers.
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$B_n,B_n\left(x\right)$	Bernoulli number and polynomial (§24.2(i)).
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19: 24.8 Series Expansions

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24.8.6 $$B_{4n+2}=(8n+4)\sum _{k=1}^{\mathrm{\infty }}\frac{k^{4n+1}}{{\mathrm{e}}^{2\pi k}-1},$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: integer and $n$: integer

Referenced by:

§24.8(ii)

Permalink:

http://dlmf.nist.gov/24.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.8(ii), §24.8 and Ch.24

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24.8.7 $$B_{2n}=\frac{{(-1)}^{n+1}4n}{2^{2n}-1}\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n-1}}{{\mathrm{e}}^{\pi k}+{(-1)}^{k+n}},$$ $n=2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $k$: integer and $n$: integer

Permalink:

http://dlmf.nist.gov/24.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.8(ii), §24.8 and Ch.24

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24.8.8 $$\frac{B_{2n}}{4n}\left({\alpha }^n-{(-\beta )}^n\right)={\alpha }^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n-1}}{{\mathrm{e}}^{2\alpha k}-1}-{(-\beta )}^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n-1}}{{\mathrm{e}}^{2\beta k}-1},$$ $n=2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\mathrm{e}$: base of natural logarithm, $k$: integer, $n$: integer, $\alpha$: parameter and $\beta$: parameter

Referenced by:

§24.8(ii)

Permalink:

http://dlmf.nist.gov/24.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.8(ii), §24.8 and Ch.24

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20: 5.15 Polygamma Functions

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5.15.8 $${\psi }^{\prime }\left(z\right)\sim \frac{1}{z}+\frac{1}{2z^2}+\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{z^{2k+1}}.$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: Poincaré asymptotic expansion, $\psi \left(z\right)$: psi (or digamma) function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.4.12

Permalink:

http://dlmf.nist.gov/5.15.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §5.15 and Ch.5

►For $B_{2k}$ see §24.2(i). …