Bernoulli monosplines

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11: 24.5 Recurrence Relations

§24.5 Recurrence Relations

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24.5.1

\sum_{k=0}^{n-1}{n\choose k}B_{k}\left(x\right)=nx^{n-1},

n=2,3,\dots

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Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex
Permalink:: http://dlmf.nist.gov/24.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §24.5(i), §24.5 and Ch.24

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24.5.3

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

n=2,3,\dots

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Symbols:: $B_{\NVar{n}}$ : Bernoulli 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.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §24.5(i), §24.5 and Ch.24

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§24.5(ii) Other Identities

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§24.5(iii) Inversion Formulas

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12: 24.21 Software

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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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13: 24.10 Arithmetic Properties

§24.10 Arithmetic Properties

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§24.10(i) Von Staudt–Clausen Theorem

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§24.10(ii) Kummer Congruences

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§24.10(iii) Voronoi’s Congruence

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§24.10(iv) Factors

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14: 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)).
$\widetilde{B}_{n}\left(x\right)$	periodic Bernoulli function $B_{n}\left(x-\left\lfloor x\right\rfloor\right)$ .
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15: 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:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : 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}|\geq|B_{2n}\left(x\right)-B_{2n}|,

1\geq x\geq 0

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : 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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24.9.4

\frac{2(2n+1)!}{(2\pi)^{2n+1}}>(-1)^{n+1}B_{2n+1}\left(x\right)>0,

n=2,3,\dots

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Symbols:: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $n$ : integer and $x$ : real or complex
Referenced by:: §24.9
Permalink:: http://dlmf.nist.gov/24.9.E4
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 e}\right)^{2n}>(-1)^{n+1}B_{2n}>4\sqrt{\pi n}% \left(\frac{n}{\pi e}\right)^{2n},

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Symbols:: $B_{\NVar{n}}$ : 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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16: 24.6 Explicit Formulas

§24.6 Explicit Formulas

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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},

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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

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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}},

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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

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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}{2k% \choose k+j}j^{2n}.

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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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24.6.9

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

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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

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17: 24.7 Integral Representations

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§24.7(i) Bernoulli and Euler Numbers

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24.7.1

B_{2n}=(-1)^{n+1}\frac{4n}{1-2^{1-2n}}\int_{0}^{\infty}\frac{t^{2n-1}}{e^{2\pi t% }+1}\,\mathrm{d}t=(-1)^{n+1}\frac{2n}{1-2^{1-2n}}\int_{0}^{\infty}t^{2n-1}e^{-% \pi t}\operatorname{sech}\left(\pi t\right)\,\mathrm{d}t,

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\int$ : integral, $n$ : integer and $t$ : real or complex
Permalink:: http://dlmf.nist.gov/24.7.E1
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}^{\infty}t^{2n}{\operatorname{% sech}}^{2}\left(\pi t\right)\,\mathrm{d}t,

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{sech}\NVar{z}$ : 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}^{\infty}t^{2n}{\operatorname{csch}}^{2}\left(\pi t% \right)\,\mathrm{d}t,

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{csch}\NVar{z}$ : 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(ii) Bernoulli and Euler Polynomials

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18: 24.15 Related Sequences of Numbers

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§24.15(i) Genocchi Numbers

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24.15.2

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

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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

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§24.15(ii) Tangent Numbers

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24.15.4

T_{2n-1}=(-1)^{n-1}\frac{2^{2n}(2^{2n}-1)}{2n}B_{2n},

n=1,2,\dots

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $n$ : integer and $T_{n}$ : tangent numbers
Referenced by:: §24.19(i)
Permalink:: http://dlmf.nist.gov/24.15.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §24.15(ii), §24.15 and Ch.24

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§24.15(iii) Stirling Numbers

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19: 24 Bernoulli and Euler Polynomials

Chapter 24 Bernoulli and Euler Polynomials

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20: Karl Dilcher

… ►Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. … ►

24 Bernoulli and Euler Polynomials

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