Bernoulli and Euler numbers and polynomials

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11: 24.4 Basic Properties

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§24.4(iv) Finite Expansions

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24.4.14 $$E_{n-1}\left(x\right)=\frac{2}{n}\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)(1-2^k)B_k{x}^{n-k},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: 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/24.4.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iv), §24.4 and Ch.24

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24.4.26 $$E_n\left(0\right)=-E_n\left(1\right)=-\frac{2}{n+1}(2^{n+1}-1)B_{n+1},$$ $n>0$.

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

A&S Ref:

23.1.20

Referenced by:

§24.9, Erratum (V1.0.5) for Equation (24.4.26)

Permalink:

http://dlmf.nist.gov/24.4.E26

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

This equation is true only for $n>0$. Previously, $n=0$ was also allowed.

Reported 2012-05-14 by Vladimir Yurovsky

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

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24.4.30 $$E_{2n-1}\left(\frac{1}{3}\right)=-E_{2n-1}\left(\frac{2}{3}\right)=-\frac{(1-3^{1-2n})(2^{2n}-1)}{2n}{B}_{2n},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

A&S Ref:

23.1.22

Permalink:

http://dlmf.nist.gov/24.4.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

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24.4.31 $$B_n\left(\frac{1}{4}\right)={(-1)}^n{B}_n\left(\frac{3}{4}\right)=-\frac{1-2^{1-n}}{2^n}{B}_n-\frac{n}{4^n}{E}_{n-1},$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $E_n$: Euler numbers and $n$: integer

A&S Ref:

23.1.22

Permalink:

http://dlmf.nist.gov/24.4.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

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12: 24.13 Integrals

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24.13.5 $${\int }_{1/4}^{3/4}{B}_n\left(t\right)dt=\frac{E_n}{2^{2n+1}}.$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $E_n$: Euler numbers, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Referenced by:

§24.13(i)

Permalink:

http://dlmf.nist.gov/24.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(i), §24.13 and Ch.24

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24.13.8 $${\int }_0^1{E}_n\left(t\right)dt=-2\frac{E_{n+1}\left(0\right)}{n+1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}{B}_{n+2},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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24.13.9 $${\int }_0^{1/2}{E}_{2n}\left(t\right)dt=-\frac{E_{2n+1}\left(0\right)}{2n+1}=\frac{2(2^{2n+2}-1)B_{2n+2}}{(2n+1)(2n+2)},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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24.13.11 $${\int }_0^1{E}_n\left(t\right)E_m\left(t\right)dt={(-1)}^{n}4\frac{(2^{m+n+2}-1)m!n!}{(m+n+2)!}{B}_{m+n+2}.$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $m$: integer, $n$: integer and $t$: real or complex

A&S Ref:

23.1.12

Referenced by:

§24.13(ii)

Permalink:

http://dlmf.nist.gov/24.13.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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13: 24.7 Integral Representations

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§24.7(i) Bernoulli and Euler Numbers

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24.7.3 $$B_{2n}={(-1)}^{n+1}\frac{\pi }{1-2^{1-2n}}{\int }_0^{\mathrm{\infty }}{t}^{2n}{\mathrm{sech}}^2\left(\pi t\right)dt,$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{sech}z$: 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 $$B_{2n}={(-1)}^{n+1}\pi {\int }_0^{\mathrm{\infty }}{t}^{2n}{\mathrm{csch}}^2\left(\pi t\right)dt,$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{csch}z$: 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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24.7.5 $$B_{2n}={(-1)}^n\frac{2n(2n-1)}{\pi }{\int }_0^{\mathrm{\infty }}{t}^{2n-2}\ln \left(1-{\mathrm{e}}^{-2\pi t}\right)dt.$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\ln z$: principal branch of logarithm function, $n$: integer and $t$: real or complex

Referenced by:

§24.7(i)

Permalink:

http://dlmf.nist.gov/24.7.E5

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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14: 24.8 Series Expansions

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§24.8(i) Fourier Series

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

§24.8(ii) Other Series

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

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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:

pMML, png, TeX

See also:

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

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24.8.9 $$E_{2n}={(-1)}^n\sum _{k=1}^{\mathrm{\infty }}\frac{k^{2n}}{\cosh \left(\frac{1}{2}\pi k\right)}-4\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{(2k+1)}^{2n}}{{\mathrm{e}}^{2\pi (2k+1)}-1},$$ $n=1,2,\mathrm{\dots }$.

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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:

Eulerpolynomials, infinite series expansions, other

Permalink:

http://dlmf.nist.gov/24.8.E9

Encodings:

pMML, png, TeX

See also:

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

15: 24.17 Mathematical Applications

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§24.17(iii) Number Theory

►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)). …

16: 24.16 Generalizations

§24.16 Generalizations

… ►When $x=0$ they reduce to the Bernoulli and Euler numbers of order $\mathrm{\ell }$ : … ►

24.16.13 $$E_n\left(x\right)=-\frac{2^{1-n}}{n+1}{B}_{n+1,{\chi }_4}(2x-1).$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $n$: integer, $x$: real or complex, $\chi (a)$: Dirichlet character and $B_{n,\chi }$: Generalized Bernoulli numbers

Referenced by:

§24.16(ii)

Permalink:

http://dlmf.nist.gov/24.16.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(ii), §24.16 and Ch.24

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§24.16(iii) Other Generalizations

►In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); $p$-adic integer order Bernoulli numbers (Adelberg (1996)); $p$-adic $q$-Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli–Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).

17: Software Index

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	Open Source													With Book						Commercial
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24.21(ii) $B_n$, $B_n\left(x\right)$, $E_n$, $E_n\left(x\right)$	✓	✓						✓		✓		a	✓	✓			✓		✓		✓	✓	✓		Derive, MuPAD
…

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18: 25.6 Integer Arguments

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§25.6(i) Function Values

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19: 25.16 Mathematical Applications

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§25.16(i) Distribution of Primes

… ►The prime number theorem (27.2.3) is equivalent to the statement … ►

§25.16(ii) Euler Sums

►Euler sums have the form … ► $H\left(s\right)$ has a simple pole with residue $\zeta \left(1-2r\right)$ ($=-B_{2r}/(2r)$) at each odd negative integer $s=1-2r$, $r=1,2,3,\mathrm{\dots }$. …

20: 5.11 Asymptotic Expansions

… ►For the Bernoulli numbers $B_{2k}$, see §24.2(i). … ►The scaled gamma function ${\mathrm{\Gamma }}^{\ast }\left(z\right)$ is defined in (5.11.3) and its main property is ${\mathrm{\Gamma }}^{\ast }\left(z\right)\sim 1$ as $z\to \mathrm{\infty }$ in the sector $|\mathrm{ph}z|\le \pi -\delta$. …For explicit formulas for $g_k$ in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of $g_k$ as $k\to \mathrm{\infty }$ see Boyd (1994) and Nemes (2015a). … ►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 $k=0,1,\mathrm{\dots }$, …