Bernoulli polynomials

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11: 24.9 Inequalities

§24.9 Inequalities

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

ⓘ

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

►

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

ⓘ

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

… ►

24.9.4 $$\frac{2(2n+1)!}{{(2\pi )}^{2n+1}}>{(-1)}^{n+1}{B}_{2n+1}\left(x\right)>0,$$ $n=2,3,\mathrm{\dots }$,

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Symbols:

$B_n\left(x\right)$: 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

…

12: 5.11 Asymptotic Expansions

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5.11.8 $$\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)\sim \left(z+h-\frac{1}{2}\right)\ln z-z+\frac{1}{2}\ln \left(2\pi \right)+\sum _{k=2}^{\mathrm{\infty }}\frac{{(-1)}^k{B}_k\left(h\right)}{k(k-1)z^{k-1}},$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\mathrm{Ln}z$: general logarithm function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

Referenced by:

§12.10(ii), (5.11.8), §5.11(i), §5.11(ii), Erratum (V1.0.10) for Equations (5.9.10), (5.9.11), (5.10.1), (5.11.1), (5.11.8), Erratum (V1.0.11) for Equation (5.11.8)

Permalink:

http://dlmf.nist.gov/5.11.E8

Encodings:

pMML, png, TeX

Generalization (effective with 1.0.11):

Originally $h$ in (5.11.8) was unnecessarily restricted to lie in the interval $[0,1]$. In fact, $h$ may lie anywhere in the complex plane.

Suggested 2015-02-28 by Nico Temme

Addition (effective with 1.0.10):

To increase the region of validity of this equation, the logarithm of the gamma function that appears on its left-hand side has been changed to $\mathrm{Ln}\mathrm{\Gamma }\left(z+h\right)$, where $\mathrm{Ln}$ is the general logarithm. Originally $\ln \mathrm{\Gamma }\left(z+h\right)$ was used, where $\ln$ is the principal branch of the logarithm.

Suggested 2015-02-13 by Philippe Spindel

See also:

Annotations for §5.11(i), §5.11(i), §5.11 and Ch.5

►where $h(\in ℂ)$ is fixed, and $B_k\left(h\right)$ is the Bernoulli polynomial defined in §24.2(i). … ►In terms of generalized Bernoulli polynomials $B_n^{(\mathrm{\ell })}\left(x\right)$ (§24.16(i)), we have for $k=0,1,\mathrm{\dots }$, ►

5.11.17 $$G_k(a,b)=\left(\genfrac{}{}{0.0pt}{}{a-b}{k}\right)B_k^{(a-b+1)}\left(a\right),$$

ⓘ

Defines:

$G_k(a,b)$: coefficients (locally)

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $k$: nonnegative integer, $a$: real or complex variable and $b$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.11.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

►

5.11.18 $$H_k(a,b)=\left(\genfrac{}{}{0.0pt}{}{a-b}{2k}\right)B_{2k}^{(a-b+1)}\left(\frac{a-b+1}{2}\right).$$

ⓘ

Defines:

$H_k(a,b)$: coefficients (locally)

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $k$: nonnegative integer, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.11(iii)

Permalink:

http://dlmf.nist.gov/5.11.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

…

13: 24.14 Sums

§24.14 Sums

►

§24.14(i) Quadratic Recurrence Relations

►

24.14.1 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k\left(x\right)B_{n-k}\left(y\right)=n(x+y-1)B_{n-1}\left(x+y\right)-(n-1)B_n\left(x+y\right),$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

Referenced by:

§24.14(i)

Permalink:

http://dlmf.nist.gov/24.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(i), §24.14 and Ch.24

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24.14.5 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(h\right)B_{n-k}\left(x\right)=2^n{B}_n\left(\frac{1}{2}(x+h)\right),$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(i), §24.14 and Ch.24

… ►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).

14: 25.11 Hurwitz Zeta Function

… ►

25.11.6 $$\zeta (s,a)=\frac{1}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $s\ne 1$, $\mathrm{\Re }s>-1$, $a>0$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1985a, (25), p. 231)

Referenced by:

Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)

Permalink:

http://dlmf.nist.gov/25.11.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Originally the integrand was incorrect because its numerator contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$.

Reported 2016-05-08 by Clemens Heuberger

See also:

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

… ►For ${\tilde{B}}_n\left(x\right)$ see §24.2(iii). … ►

25.11.14 $$\zeta (-n,a)=-\frac{B_{n+1}\left(a\right)}{n+1},$$ $n=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\zeta (s,a)$: Hurwitz zeta function, $n$: nonnegative integer and $a$: real or complex parameter

Keywords:

special value

Source:

Apostol (1976, (17), p. 264)

Permalink:

http://dlmf.nist.gov/25.11.E14

Encodings:

pMML, png, TeX

See also:

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

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25.11.19 $${\zeta }^{\prime }(s,a)=-\frac{\ln a}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{{(s-1)}^2}+\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2)\ln \left(x+a\right)}{{(x+a)}^{s+2}}dx-\frac{(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{{\tilde{B}}_2\left(x\right)-B_2}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $k$: integer

Keywords:

derivative

Source:

Apostol (1985a, fourth equation, p. 231); with $k=1$, ${\phi }_2(x)=\frac{1}{2}({\tilde{B}}_2\left(x\right)-B_2)$

Referenced by:

Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)

Permalink:

http://dlmf.nist.gov/25.11.E19

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Originally both integrands were incorrect because their numerators contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$.

Reported 2016-06-27 by Gergő Nemes

See also:

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

►

25.11.20 $${(-1)}^k{\zeta }^{(k)}(s,a)=\frac{{(\ln a)}^k}{a^s}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum _{r=0}^{k-1}\frac{{(\ln a)}^r}{r!{(s-1)}^{k-r+1}}-\frac{s(s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^k}{{(x+a)}^{s+2}}dx+\frac{k(2s+1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-1}}{{(x+a)}^{s+2}}dx-\frac{k(k-1)}{2}{\int }_0^{\mathrm{\infty }}\frac{({\tilde{B}}_2\left(x\right)-B_2){(\ln \left(x+a\right))}^{k-2}}{{(x+a)}^{s+2}}dx,$$ $\mathrm{\Re }s>-1$, $s\ne 1$, $a>0$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $k$: integer

Keywords:

derivative

Source:

Apostol (1985a, fourth equation, p. 231); with ${\phi }_2(x)=\frac{1}{2}({\tilde{B}}_2\left(x\right)-B_2)$

Referenced by:

Erratum (V1.0.12) for Equations (25.11.6), (25.11.19), and (25.11.20)

Permalink:

http://dlmf.nist.gov/25.11.E20

Encodings:

pMML, png, TeX

Errata (effective with 1.0.12):

Originally all three integrands were incorrect because their numerators contained the function ${\tilde{B}}_2\left(x\right)$. The correct function is $\frac{{\tilde{B}}_2\left(x\right)-B_2}{2}$.

Reported 2016-06-27 by Gergő Nemes

See also:

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

…

15: 24.8 Series Expansions

… ►

§24.8(i) Fourier Series

… ►

24.8.1 $$B_{2n}\left(x\right)={(-1)}^{n+1}\frac{2(2n)!}{{(2\pi )}^{2n}}\sum _{k=1}^{\mathrm{\infty }}\frac{\cos \left(2\pi kx\right)}{k^{2n}},$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $!$: factorial (as in $n!$), $k$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.18

Referenced by:

§24.8(i)

Permalink:

http://dlmf.nist.gov/24.8.E1

Encodings:

pMML, png, TeX

See also:

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

►

24.8.2 $$B_{2n+1}\left(x\right)={(-1)}^{n+1}\frac{2(2n+1)!}{{(2\pi )}^{2n+1}}\sum _{k=1}^{\mathrm{\infty }}\frac{\sin \left(2\pi kx\right)}{k^{2n+1}}.$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\sin z$: sine function, $k$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.17

Permalink:

http://dlmf.nist.gov/24.8.E2

Encodings:

pMML, png, TeX

See also:

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

… ►

24.8.3 $$B_n\left(x\right)=-\frac{n!}{{(2\pi \mathrm{i})}^n}\sum _{\begin{array}{c}\hfill k=-\mathrm{\infty }\hfill \\ \hfill k\ne 0\hfill \end{array}}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}kx}}{k^n}.$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $k$: integer, $n$: integer and $x$: real or complex

Referenced by:

§24.8(i)

Permalink:

http://dlmf.nist.gov/24.8.E3

Encodings:

pMML, png, TeX

See also:

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

… ►

§24.8(ii) Other Series

…

16: 24.7 Integral Representations

… ►

§24.7(ii) Bernoulli and Euler Polynomials

… ►

24.7.7 $$B_{2n}\left(x\right)={(-1)}^{n+1}2n{\int }_0^{\mathrm{\infty }}\frac{\cos \left(2\pi x\right)-{\mathrm{e}}^{-2\pi t}}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n-1}dt,$$ $n=1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\int$: integral, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E7

Encodings:

pMML, png, TeX

See also:

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

►

24.7.8 $$B_{2n+1}\left(x\right)={(-1)}^{n+1}(2n+1){\int }_0^{\mathrm{\infty }}\frac{\sin \left(2\pi x\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n}dt.$$

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E8

Encodings:

pMML, png, TeX

See also:

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

… ►

24.7.11 $$B_n\left(x\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-c-\mathrm{i}\mathrm{\infty }}^{-c+\mathrm{i}\mathrm{\infty }}{(x+t)}^n{\left(\frac{\pi }{\sin \left(\pi t\right)}\right)}^{2}dt,$$ .

ⓘ

Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $n$: integer, $t$: real or complex and $x$: real or complex

Keywords:

Mellin transform

Referenced by:

§24.7(ii)

Permalink:

http://dlmf.nist.gov/24.7.E11

Encodings:

pMML, png, TeX

See also:

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

…

17: 24.12 Zeros

… ►

§24.12(i) Bernoulli Polynomials: Real Zeros

►In the interval $0\le x\le 1$ the only zeros of $B_{2n+1}\left(x\right)$, $n=1,2,\mathrm{\dots }$, are $0,\frac{1}{2},1$, and the only zeros of $B_{2n}\left(x\right)-B_{2n}$, $n=1,2,\mathrm{\dots }$, are $0,1$. ►For the interval denote the zeros of $B_n\left(x\right)$ by $x_j^{(n)}$, $j=1,2,\mathrm{\dots }$, with … ►Let $R(n)$ be the total number of real zeros of $B_n\left(x\right)$. … ► $B_n\left(x\right)$, $n=1,2,\mathrm{\dots }$, has no multiple zeros. …

18: 25.2 Definition and Expansions

… ►

25.2.9 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-\frac{1}{2}{N}^{-s}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}{N}^{1-s-2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_N^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$; $n,N=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Derivable from (25.2.8) by repeated integration by parts.

Referenced by:

§25.18(i), (25.2.10), §25.2(iii)

Permalink:

http://dlmf.nist.gov/25.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

►

25.2.10 $$\zeta \left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$, $n=1,2,3,\mathrm{\dots }$.

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula

Source:

Derivable from (25.2.9) with $N=1$.

A&S Ref:

23.2.3

Permalink:

http://dlmf.nist.gov/25.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

►For $B_{2k}$ see §24.2(i), and for ${\tilde{B}}_n\left(x\right)$ see §24.2(iii). …

19: 17.3 $q$-Elementary and $q$-Special Functions

… ►

§17.3(iii) Bernoulli Polynomials; Euler and Stirling Numbers

►

$q$-Bernoulli Polynomials

►

17.3.7 $${\beta }_n(x,q)={(1-q)}^{1-n}\sum _{r=0}^n{(-1)}^r\left(\genfrac{}{}{0.0pt}{}{n}{r}\right)\frac{r+1}{(1-q^{r+1})}{q}^{rx}.$$

ⓘ

Defines:

${\beta }_n(x,q)$: $q$-Bernoulli polynomial

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $q$: complex base, $n$: nonnegative integer, $r$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.3.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §17.3(iii), §17.3(iii), §17.3 and Ch.17

… ►The ${\beta }_n(x,q)$ are, in fact, rational functions of $q$, and not necessarily polynomials. …

20: 25.16 Mathematical Applications

… ►

25.16.6 $$H\left(s\right)=-{\zeta }^{\prime }\left(s\right)+\gamma \zeta \left(s\right)+\frac{1}{2}\zeta \left(s+1\right)+\sum _{r=1}^k\zeta \left(1-2r\right)\zeta \left(s+2r\right)+\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{2k+2}}dx,$$

ⓘ

Symbols:

$\gamma$: Euler’s constant, $H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, series representation

Source:

Apostol and Vu (1984, (3), p. 87)

Permalink:

http://dlmf.nist.gov/25.16.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

►

25.16.7 $$H\left(s\right)=\frac{1}{2}\zeta \left(s+1\right)+\frac{\zeta \left(s\right)}{s-1}-\sum _{r=1}^k\left(\genfrac{}{}{0.0pt}{}{s+2r-2}{2r-1}\right)\zeta \left(1-2r\right)\zeta \left(s+2r\right)-\left(\genfrac{}{}{0.0pt}{}{s+2k}{2k+1}\right)\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}{\int }_n^{\mathrm{\infty }}\frac{{\tilde{B}}_{2k+1}\left(x\right)}{x^{s+2k+1}}dx.$$

ⓘ

Symbols:

$H\left(s\right)$: Euler sums, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler sum, finite sum, improper integral, infinite series, integral representation, series representation

Source:

Apostol and Vu (1984, (6), p. 88)

Permalink:

http://dlmf.nist.gov/25.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §25.16(ii), §25.16 and Ch.25

…