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DLMF: §24.8 Series Expansions ‣ Properties ‣ Chapter 24 Bernoulli and Euler Polynomials

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24 Bernoulli and Euler Polynomials Properties 24.7 Integral Representations 24.9 Inequalities

§24.8 Series Expansions

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Permalink:: http://dlmf.nist.gov/24.8
See also:: Annotations for Ch.24

Contents

§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 $n=1,2,\dots$ and $0\leq x\leq 1$ , then

24.8.1		$\displaystyle B_{2n}\left(x\right)$	$\displaystyle=(-1)^{n+1}\frac{2(2n)!}{(2\pi)^{2n}}\sum_{k=1}^{\infty}\frac{% \cos\left(2\pi kx\right)}{k^{2n}},$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : 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		$\displaystyle B_{2n+1}\left(x\right)$	$\displaystyle=(-1)^{n+1}\frac{2(2n+1)!}{(2\pi)^{2n+1}}\sum_{k=1}^{\infty}\frac% {\sin\left(2\pi kx\right)}{k^{2n+1}}.$
ⓘ 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 $n!$ ), $\sin\NVar{z}$ : 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 $0<x<1$ .

If $n=1$ with $0<x<1$ , or $n=2,3,\dots$ with $0\leq x\leq 1$ , then

24.8.3		$B_{n}\left(x\right)=-\frac{n!}{(2\pi i)^{n}}\sum_{\begin{subarray}{c}k=-\infty% \\ k\neq 0\end{subarray}}^{\infty}\frac{e^{2\pi ikx}}{k^{n}}.$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : 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 $n=1,2,\dots$ and $0\leq x\leq 1$ , then

24.8.4		$\displaystyle E_{2n}\left(x\right)$	$\displaystyle=(-1)^{n}\frac{4(2n)!}{\pi^{2n+1}}\sum_{k=0}^{\infty}\frac{\sin% \left((2k+1)\pi x\right)}{(2k+1)^{2n+1}},$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\sin\NVar{z}$ : 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		$\displaystyle E_{2n-1}\left(x\right)$	$\displaystyle=(-1)^{n}\frac{4(2n-1)!}{\pi^{2n}}\sum_{k=0}^{\infty}\frac{\cos% \left((2k+1)\pi x\right)}{(2k+1)^{2n}}.$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : 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		$\displaystyle B_{4n+2}$	$\displaystyle=(8n+4)\sum_{k=1}^{\infty}\frac{k^{4n+1}}{e^{2\pi k}-1},$
$n=1,2,\dots$ ,
ⓘ Symbols: $B_{\NVar{n}}$ : 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		$\displaystyle B_{2n}$	$\displaystyle=\frac{(-1)^{n+1}4n}{2^{2n}-1}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{% e^{\pi k}+(-1)^{k+n}},$
$n=2,3,\dots$ .
ⓘ Symbols: $B_{\NVar{n}}$ : 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		$\frac{B_{2n}}{4n}\left(\alpha^{n}-(-\beta)^{n}\right)=\alpha^{n}\sum_{k=1}^{% \infty}\frac{k^{2n-1}}{e^{2\alpha k}-1}-(-\beta)^{n}\sum_{k=1}^{\infty}\frac{k% ^{2n-1}}{e^{2\beta k}-1},$
$n=2,3,\dots$ .
ⓘ Symbols: $B_{\NVar{n}}$ : 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		$E_{2n}=(-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{\cosh\left(\tfrac{1}{2}\pi k% \right)}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^{2n}}{e^{2\pi(2k+1)}-1},$
$n=1,2,\dots$ .
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : 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 Representations 24.9 Inequalities

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