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11: 24.19 Methods of Computation

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§24.19(i) Bernoulli and Euler Numbers and Polynomials

… ►For algorithms for computing $B_n$, $E_n$, $B_n\left(x\right)$, and $E_n\left(x\right)$ see Spanier and Oldham (1987, pp. 37, 41, 171, and 179–180). ►

§24.19(ii) Values of $B_n$ Modulo $p$

… ►We list here three methods, arranged in increasing order of efficiency. ►

Tanner and Wagstaff (1987) derives a congruence $(modp)$ for Bernoulli numbers in terms of sums of powers. See also §24.10(iii).

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12: 24.9 Inequalities

§24.9 Inequalities

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

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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

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

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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

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24.9.3 $$4^{-n}|E_{2n}|>{(-1)}^n{E}_{2n}\left(x\right)>0,$$

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Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials, $n$: integer and $x$: real or complex

A&S Ref:

23.1.13

Referenced by:

§24.9, §24.9

Permalink:

http://dlmf.nist.gov/24.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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24.9.5 $$\frac{4(2n-1)!}{{\pi }^{2n}}\frac{2^{2n}-1}{2^{2n}-2}>{(-1)}^n{E}_{2n-1}\left(x\right)>0.$$

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Symbols:

$E_n\left(x\right)$: Euler polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $x$: real or complex

A&S Ref:

23.1.14

Referenced by:

§24.9, §24.9

Permalink:

http://dlmf.nist.gov/24.9.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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13: 24.5 Recurrence Relations

§24.5 Recurrence Relations

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24.5.1 $$\sum _{k=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k\left(x\right)=n{x}^{n-1},$$ $n=2,3,\mathrm{\dots }$,

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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

Permalink:

http://dlmf.nist.gov/24.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

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24.5.2 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(x\right)+E_n\left(x\right)=2x^n,$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$E_n\left(x\right)$: 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/24.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(i), §24.5 and Ch.24

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§24.5(ii) Other Identities

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§24.5(iii) Inversion Formulas

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14: 24 Bernoulli and Euler Polynomials

Chapter 24 Bernoulli and Euler Polynomials

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15: 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(ii) Bernoulli and Euler Polynomials

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24.7.9 $$E_{2n}\left(x\right)={(-1)}^{n}4{\int }_0^{\mathrm{\infty }}\frac{\sin \left(\pi x\right)\cosh \left(\pi t\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n}dt,$$

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Symbols:

$E_n\left(x\right)$: Euler 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.E9

Encodings:

pMML, png, TeX

See also:

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

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24.7.10 $$E_{2n+1}\left(x\right)={(-1)}^{n+1}4{\int }_0^{\mathrm{\infty }}\frac{\cos \left(\pi x\right)\sinh \left(\pi t\right)}{\cosh \left(2\pi t\right)-\cos \left(2\pi x\right)}{t}^{2n+1}dt.$$

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Symbols:

$E_n\left(x\right)$: Euler 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, $\sinh z$: hyperbolic sine function, $\int$: integral, $n$: integer, $t$: real or complex and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.7.E10

Encodings:

pMML, png, TeX

See also:

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

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16: 24.12 Zeros

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§24.12(i) Bernoulli Polynomials: Real Zeros

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§24.12(iii) Complex Zeros

►For complex zeros of Bernoulli and Euler polynomials, see Delange (1987) and Dilcher (1988). A related topic is the irreducibility of Bernoulli and Euler polynomials. … ►

§24.12(iv) Multiple Zeros

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17: 24.6 Explicit Formulas

§24.6 Explicit Formulas

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24.6.1 $$B_{2n}=\sum _{k=2}^{2n+1}\frac{{(-1)}^{k-1}}{k}\left(\genfrac{}{}{0pt}{}{2n+1}{k}\right)\sum _{j=1}^{k-1}{j}^{2n},$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.7 $$B_n\left(x\right)=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(x+j)}^n,$$

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Symbols:

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

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.8 $$E_n\left(x\right)=\frac{1}{2^n}\sum _{k=1}^{n+1}\sum _{j=0}^{k-1}{(-1)}^j\left(\genfrac{}{}{0pt}{}{n+1}{k}\right){(x+j)}^n.$$

ⓘ

Symbols:

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

Referenced by:

§24.6, §24.6

Permalink:

http://dlmf.nist.gov/24.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.9 $$B_n=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right)j^n,$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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18: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

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24.11.1 $${(-1)}^{n+1}{B}_{2n}\sim \frac{2(2n)!}{{(2\pi )}^{2n}},$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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24.11.2 $${(-1)}^{n+1}{B}_{2n}\sim 4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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24.11.4 $${(-1)}^n{E}_{2n}\sim 8\sqrt{\frac{n}{\pi }}{\left(\frac{4n}{\pi \mathrm{e}}\right)}^{2n}.$$

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Symbols:

$E_n$: Euler numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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24.11.6

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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 $n!$), $\lfloor x\rfloor$: floor of $x$ , $\sin z$: sine function, $n$: integer and $x$: real or complex

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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19: Karl Dilcher

… ►Over the years he authored or coauthored numerous papers on Bernoulli numbers and related topics, and he maintains a large on-line bibliography on the subject. … ►

24 Bernoulli and Euler Polynomials

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20: 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 ►

24.8.4 $$E_{2n}\left(x\right)={(-1)}^n\frac{4(2n)!}{{\pi }^{2n+1}}\sum _{k=0}^{\mathrm{\infty }}\frac{\sin \left((2k+1)\pi x\right)}{{(2k+1)}^{2n+1}},$$

ⓘ

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:

pMML, png, TeX

See also:

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

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24.8.5 $$E_{2n-1}\left(x\right)={(-1)}^n\frac{4(2n-1)!}{{\pi }^{2n}}\sum _{k=0}^{\mathrm{\infty }}\frac{\cos \left((2k+1)\pi x\right)}{{(2k+1)}^{2n}}.$$

ⓘ

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:

pMML, png, TeX

See also:

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

►

§24.8(ii) Other Series

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