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11: 24.4 Basic Properties

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§24.4(iv) Finite Expansions

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24.4.31 $$B_n\left(\frac{1}{4}\right)={(-1)}^n{B}_n\left(\frac{3}{4}\right)=-\frac{1-2^{1-n}}{2^n}{B}_n-\frac{n}{4^n}{E}_{n-1},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

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

A&S Ref:

23.1.22

Permalink:

http://dlmf.nist.gov/24.4.E31

Encodings:

pMML, png, TeX

See also:

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

… ►Let $P(x)$ denote any polynomial in $x$, and after expanding set ${(B(x))}^n=B_n\left(x\right)$ and ${(E(x))}^n=E_n\left(x\right)$. … ►

§24.4(ix) Relations to Other Functions

►For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).

12: Foreword

… ► The success of the original handbook, widely referred to as “Abramowitz and Stegun” (“A&S”), derived not only from the fact that it provided critically useful scientific data in a highly accessible format, but also because it served to standardize definitions and notations for special functions. … ►Particular attention is called to the generous support of the National Science Foundation, which made possible the participation of experts from academia and research institutes worldwide. …

13: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

… ►The denominator of $B_{2n}$ is the product of all these primes $p$. … ►

§24.10(ii) Kummer Congruences

… ►Let $B_{2n}=N_{2n}/D_{2n}$, with $N_{2n}$ and $D_{2n}$ relatively prime and $D_{2n}>0$. … ►

§24.10(iv) Factors

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14: Viewing DLMF Interactive 3D Graphics

… ►WebGL is supported in the current versions of most common web browsers. … ►1, some advanced features of X3DOM are currently not fully supported (see x3dom.org). …If you have trouble viewing the WebGL visualizations in your web browser, see x3dom.org or caniuse.com/webgl for information on WebGL browser support. … ►Please see caniuse.com/webgl or x3dom.org for information on WebGL browser support.

15: 35 Functions of Matrix Argument

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16: Gloria Wiersma

… ► 1937 in Washington, DC) joined the NIST staff in 1973, where she occupied various positions providing support for the Physics Laboratory until 1993. …

17: 24.5 Recurrence Relations

§24.5 Recurrence Relations

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§24.5(ii) Other Identities

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24.5.6 $$\sum _{k=2}^n\left(\genfrac{}{}{0pt}{}{n}{k-2}\right)\frac{B_k}{k}=\frac{1}{(n+1)(n+2)}-B_{n+1},$$ $n=2,3,\mathrm{\dots }$,

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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

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24.5.8 $$\sum _{k=0}^n\frac{2^{2k}{B}_{2k}}{(2k)!(2n+1-2k)!}=\frac{1}{(2n)!},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$B_n$: 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:

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See also:

Annotations for §24.5(ii), §24.5 and Ch.24

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§24.5(iii) Inversion Formulas

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18: Bibliography Y

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Simple Variational Proof That Any Two-Dimensional Potential Well Supports at Least One Bound State. American Journal of Physics 57 (1), pp. 85–86.

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External Links:

Document

Referenced by:

§1.18(viii).

Encodings:

BibTeX

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The calculation of the Riemann zeta function in the complex domain. USSR Comput. Math. and Math. Phys. 25 (2), pp. 111–119.

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External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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Coulomb wave functions in repulsive fields. Phys. Rev. 49 (2), pp. 174–189.

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External Links:

Document, ZentralBlatt

Referenced by:

§33.10(i), §33.2(ii), §33.2(iii), §33.2(iv), §33.5(i), §33.5(iv), §33.9(ii).

Encodings:

BibTeX

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19: 24.6 Explicit Formulas

§24.6 Explicit Formulas

►The identities in this section hold for $n=1,2,\mathrm{\dots }$. … ►

24.6.1 $$B_{2n}=\sum _{k=2}^{2n+1}\frac{{(-1)}^{k-1}}{k}\left(\genfrac{}{}{0pt}{}{2n+1}{k}\right)\sum _{j=1}^{k-1}{j}^{2n},$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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

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24.6.3 $$B_{2n}=\sum _{k=1}^n\frac{(k-1)!k!}{(2k+1)!}\sum _{j=1}^k{(-1)}^{j-1}\left(\genfrac{}{}{0pt}{}{2k}{k+j}\right)j^{2n}.$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.11 $$B_n=\frac{n}{2^n(2^n-1)}\sum _{k=1}^n\sum _{j=0}^{k-1}{(-1)}^{j+1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)j^{n-1},$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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

…

20: 24.9 Inequalities

§24.9 Inequalities

►Except where otherwise noted, the inequalities in this section hold for $n=1,2,\mathrm{\dots }$. … ►

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

►(24.9.3)–(24.9.5) hold for $\frac{1}{2}>x>0$. … ►(24.9.6)–(24.9.7) hold for $n=2,3,\mathrm{\dots }$. …