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

§24.8(i) Fourier Series
§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 $n=1,2,\dots$ and $0\leq x\leq 1$ , then

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

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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, $\cos\NVar{z}$ : cosine function, : factorial (as in ), : integer, : integer and : 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 $B_{2n+1}\left(x\right)=(-1)^{n+1}\frac{2(2n+1)!}{(2\pi)^{2n+1}}\sum_{k=1}^{% \infty}\frac{\sin\left(2\pi kx\right)}{k^{2n+1}}.$

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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 ), $\sin\NVar{z}$ : sine function, : integer, : integer and : 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 and .

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

24.8.3 $B_{n}\left(x\right)=-\frac{n!}{(2\pi i)^{n}}\sum_{\begin{subarray}{c}k=-\infty% \\ k\neq 0\end{subarray}}^{\infty}\frac{e^{2\pi ikx}}{k^{n}}.$

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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, $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $\mathrm{i}$ : imaginary unit, : integer, : integer and : 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 $E_{2n}\left(x\right)=(-1)^{n}\frac{4(2n)!}{\pi^{2n+1}}\sum_{k=0}^{\infty}\frac% {\sin\left((2k+1)\pi x\right)}{(2k+1)^{2n+1}},$

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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 ), $\sin\NVar{z}$ : sine function, : integer, : integer and : 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 $E_{2n-1}\left(x\right)=(-1)^{n}\frac{4(2n-1)!}{\pi^{2n}}\sum_{k=0}^{\infty}% \frac{\cos\left((2k+1)\pi x\right)}{(2k+1)^{2n}}.$

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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 ), : integer, : integer and : 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 is odd.
Permalink:: http://dlmf.nist.gov/24.8.ii
See also:: Annotations for §24.8 and Ch.24

24.8.6 $B_{4n+2}=(8n+4)\sum_{k=1}^{\infty}\frac{k^{4n+1}}{e^{2\pi k}-1},$ $n=1,2,\dots$ ,

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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, : integer and : 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 $B_{2n}=\frac{(-1)^{n+1}4n}{2^{2n}-1}\sum_{k=1}^{\infty}\frac{k^{2n-1}}{e^{\pi k% }+(-1)^{k+n}},$ $n=2,3,\dots$ .

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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, : integer and : 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 $\frac{B_{2n}}{4n}\left(\alpha^{n}-(-\beta)^{n}\right)=\alpha^{n}\sum_{k=1}^{% \infty}\frac{k^{2n-1}}{e^{2\alpha k}-1}-(-\beta)^{n}\sum_{k=1}^{\infty}\frac{k% ^{2n-1}}{e^{2\beta k}-1},$ $n=2,3,\dots$ .

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Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\mathrm{e}$ : base of natural logarithm, : integer, : 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 $E_{2n}=(-1)^{n}\sum_{k=1}^{\infty}\frac{k^{2n}}{\cosh\left(\tfrac{1}{2}\pi k% \right)}-4\sum_{k=0}^{\infty}\frac{(-1)^{k}(2k+1)^{2n}}{e^{2\pi(2k+1)}-1},$ $n=1,2,\dots$ .

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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, : integer and : 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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§24.8 Series Expansions

Contents

§24.8(i) Fourier Series

§24.8(ii) Other Series