Euler polynomials

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11: 24.12 Zeros

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§24.12(ii) Euler 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). … ►

§24.12(iv) Multiple Zeros

… ►The only polynomial $E_n\left(x\right)$ with multiple zeros is $E_5\left(x\right)=(x-\frac{1}{2}){(x^2-x-1)}^2$.

12: 24.9 Inequalities

§24.9 Inequalities

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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 Bernoulli and Euler Polynomials

Chapter 24 Bernoulli and Euler Polynomials

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14: 24.7 Integral Representations

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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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15: 24.5 Recurrence Relations

§24.5 Recurrence Relations

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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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16: 24.8 Series Expansions

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

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

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

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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 $n!$), $k$: 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

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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

17: 24.11 Asymptotic Approximations

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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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18: 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). …

19: 24.6 Explicit Formulas

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24.6.6 $$E_{2n}=\sum _{k=1}^{2n}\frac{{(-1)}^k}{2^{k-1}}\left(\genfrac{}{}{0pt}{}{2n+1}{k+1}\right)\sum _{j=0}^{\lfloor \frac{1}{2}k-\frac{1}{2}\rfloor }\left(\genfrac{}{}{0pt}{}{k}{j}\right){(k-2j)}^{2n}.$$

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Symbols:

$E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$ , $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E6

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

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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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20: Karl Dilcher

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24 Bernoulli and Euler Polynomials

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