24 Bernoulli and Euler PolynomialsProperties24.7 Integral Representations24.9 Inequalities

§24.8 Series Expansions

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http://dlmf.nist.gov/24.8

See also:

Annotations for Ch.24

Contents

1. §24.8(i) Fourier Series
2. §24.8(ii) Other Series

§24.8(i) Fourier Series

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Keywords:

Bernoulli polynomials, Euler polynomials, Fourier, infinite series expansions

Notes:

For (24.8.1)–(24.8.3) see Apostol (1976, p. 267). For (24.8.4) and (24.8.5) see Erdélyi et al. (1953a, p. 42).

Permalink:

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

See also:

Annotations for §24.8 and Ch.24

If $n=1,2,\mathrm{\dots }$ and $0\le x\le 1$, then

24.8.1		$B_{2n}\left(x\right)$	$={(-1)}^{n+1}{\displaystyle \frac{2(2n)!}{{(2\pi )}^{2n}}}{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{\displaystyle \frac{\cos \left(2\pi kx\right)}{k^{2n}}},$
ⓘ 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: TeX, pMML, png See also: Annotations for §24.8(i), §24.8 and Ch.24
24.8.2		$B_{2n+1}\left(x\right)$	$={(-1)}^{n+1}{\displaystyle \frac{2(2n+1)!}{{(2\pi )}^{2n+1}}}{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{\displaystyle \frac{\sin \left(2\pi kx\right)}{k^{2n+1}}}.$
ⓘ 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: TeX, pMML, png See also: Annotations for §24.8(i), §24.8 and Ch.24

The second expansion holds also for $n=0$ and .

If $n=1$ with , or $n=2,3,\mathrm{\dots }$ with $0\le x\le 1$, then

24.8.3		$$B_n\left(x\right)=-\frac{n!}{{(2\pi \mathrm{i})}^n}\sum _{\begin{array}{c}k=-\mathrm{\infty }\\ k\ne 0\end{array}}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{2\pi \mathrm{i}kx}}{k^n}.$$
ⓘ 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: TeX, pMML, png See also: Annotations for §24.8(i), §24.8 and Ch.24

If $n=1,2,\mathrm{\dots }$ and $0\le x\le 1$, then

24.8.4		$E_{2n}\left(x\right)$	$={(-1)}^n{\displaystyle \frac{4(2n)!}{{\pi }^{2n+1}}}{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{\sin \left((2k+1)\pi x\right)}{{(2k+1)}^{2n+1}}},$
ⓘ Symbols: $E_n\left(x\right)$: Euler 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: 23.1.18 Referenced by: §24.8(i), §24.9 Permalink: http://dlmf.nist.gov/24.8.E4 Encodings: TeX, pMML, png See also: Annotations for §24.8(i), §24.8 and Ch.24
24.8.5		$E_{2n-1}\left(x\right)$	$={(-1)}^n{\displaystyle \frac{4(2n-1)!}{{\pi }^{2n}}}{\displaystyle \sum _{k=0}^{\mathrm{\infty }}}{\displaystyle \frac{\cos \left((2k+1)\pi x\right)}{{(2k+1)}^{2n}}}.$
ⓘ Symbols: $E_n\left(x\right)$: Euler 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: 23.1.17 Referenced by: §24.8(i) Permalink: http://dlmf.nist.gov/24.8.E5 Encodings: TeX, pMML, png See also: Annotations for §24.8(i), §24.8 and Ch.24

§24.8(ii) Other Series

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Keywords:

Bernoulli polynomials, infinite series expansions, other

Notes:

See Berndt (1975b, pp. 176–178). (24.8.8) reduces to (24.8.6) when $\alpha =\beta =\pi$ and $n$ is odd.

Permalink:

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

See also:

Annotations for §24.8 and Ch.24

24.8.6		$B_{4n+2}$	$=(8n+4){\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{\displaystyle \frac{k^{4n+1}}{{\mathrm{e}}^{2\pi k}-1}},$
$n=1,2,\mathrm{\dots }$,
ⓘ 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 Referenced by: §24.8(ii) Permalink: http://dlmf.nist.gov/24.8.E6 Encodings: TeX, pMML, png See also: Annotations for §24.8(ii), §24.8 and Ch.24
24.8.7		$B_{2n}$	$={\displaystyle \frac{{(-1)}^{n+1}4n}{2^{2n}-1}}{\displaystyle \sum _{k=1}^{\mathrm{\infty }}}{\displaystyle \frac{k^{2n-1}}{{\mathrm{e}}^{\pi k}+{(-1)}^{k+n}}},$
$n=2,3,\mathrm{\dots }$.
ⓘ 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: TeX, pMML, png See also: Annotations for §24.8(ii), §24.8 and Ch.24

Let $\alpha \beta ={\pi }^2$. Then

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: TeX, pMML, png See also: Annotations for §24.8(ii), §24.8 and Ch.24

24.8.9		$$E_{2n}={(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n}}{\cosh \left(\frac{1}{2}\pi k\right)}-4\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(2k+1)}^{2n}}{{\mathrm{e}}^{2\pi (2k+1)}-1},$$
$n=1,2,\mathrm{\dots }$.
ⓘ Symbols: $E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $k$: integer and $n$: integer Keywords: Euler polynomials, infinite series expansions, other Permalink: http://dlmf.nist.gov/24.8.E9 Encodings: TeX, pMML, png See also: Annotations for §24.8(ii), §24.8 and Ch.24

24.7 Integral Representations24.9 Inequalities

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