Euler polynomials

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1: 24.1 Special Notation

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Euler Numbers and Polynomials

… ►The notations $E_n$, $E_n\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

2: 24.3 Graphs

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Figure 24.3.2: Euler polynomials $E_n\left(x\right)$, $n=2,3,\mathrm{\dots },6$. Magnify

3: 24.18 Physical Applications

§24.18 Physical Applications

►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). ►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).

4: 24.4 Basic Properties

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24.4.2 $$E_n\left(x+1\right)+E_n\left(x\right)=2x^n.$$

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Symbols:

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

A&S Ref:

23.1.6

Permalink:

http://dlmf.nist.gov/24.4.E2

Encodings:

pMML, png, TeX

See also:

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

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§24.4(ii) Symmetry

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24.4.4 $$E_n\left(1-x\right)={(-1)}^n{E}_n\left(x\right).$$

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Symbols:

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

A&S Ref:

23.1.8

Permalink:

http://dlmf.nist.gov/24.4.E4

Encodings:

pMML, png, TeX

See also:

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

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24.4.6 $${(-1)}^{n+1}{E}_n\left(-x\right)=E_n\left(x\right)-2x^n.$$

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Symbols:

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

A&S Ref:

23.1.9

Permalink:

http://dlmf.nist.gov/24.4.E6

Encodings:

pMML, png, TeX

See also:

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

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5: 24.21 Software

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§24.21(ii) $B_n$, $B_n\left(x\right)$, $E_n$, and $E_n\left(x\right)$

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6: 24.13 Integrals

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24.13.3 $${\int }_x^{x+(1/2)}{B}_n\left(t\right)dt=\frac{E_n\left(2x\right)}{2^{n+1}},$$

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Symbols:

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

Referenced by:

§24.13(i)

Permalink:

http://dlmf.nist.gov/24.13.E3

Encodings:

pMML, png, TeX

See also:

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

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§24.13(ii) Euler Polynomials

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24.13.7 $$\int E_n\left(t\right)dt=\frac{E_{n+1}\left(t\right)}{n+1}+\text{const.},$$

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Symbols:

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

A&S Ref:

23.1.11

Referenced by:

§24.13(ii)

Permalink:

http://dlmf.nist.gov/24.13.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §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(iii) Compendia

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7: 24.16 Generalizations

§24.16 Generalizations

… ►For $\mathrm{\ell }=0,1,2,\mathrm{\dots }$, Bernoulli and Euler polynomials of order $\mathrm{\ell }$ are defined respectively by …When $x=0$ they reduce to the Bernoulli and Euler numbers of order $\mathrm{\ell }$ : … ►

$E_n^{(\mathrm{\ell })}=E_n^{(\mathrm{\ell })}\left(0\right).$

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§24.16(iii) Other Generalizations

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8: 24.2 Definitions and Generating Functions

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${\tilde{E}}_n\left(x\right)=E_n\left(x\right)$ , ,

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${\tilde{E}}_n\left(x+1\right)=-{\tilde{E}}_n\left(x\right)$ , $x\in ℝ$.

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Table 24.2.2: Bernoulli and Euler polynomials.

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$n$	$B_n\left(x\right)$	$E_n\left(x\right)$
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9: 24.17 Mathematical Applications

§24.17 Mathematical Applications

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24.17.2 $$R_m(n)=\frac{1}{2(m-1)!}{\int }_a^n{f}^{(m)}(x){\tilde{E}}_{m-1}\left(h-x\right)dx.$$

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Symbols:

$dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, ${\tilde{E}}_n\left(x\right)$: periodic Euler functions, $m$: integer, $x$: real or complex, $h$: parameter, $a$: integer, $n$: integer, $f^{(m)}(x)$: integrable function and $R_m(n)$: remainder

Permalink:

http://dlmf.nist.gov/24.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(i), §24.17(i), §24.17 and Ch.24

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24.17.3 $$S_n(x)=\frac{{\tilde{E}}_n\left(x+\frac{1}{2}n+\frac{1}{2}\right)}{{\tilde{E}}_n\left(\frac{1}{2}n+\frac{1}{2}\right)},$$ $n=0,1,\mathrm{\dots }$,

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Symbols:

${\tilde{E}}_n\left(x\right)$: periodic Euler functions, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.17(ii), §24.17(ii), §24.17 and Ch.24

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§24.17(iii) Number Theory

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10: 24.14 Sums

§24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

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24.14.3 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(h\right)E_{n-k}\left(x\right)=2(E_{n+1}\left(x+h\right)-(x+h-1)E_n\left(x+h\right)),$$

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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/24.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(i), §24.14 and Ch.24

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24.14.4 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k{E}_{n-k}=-2^{n+1}{E}_{n+1}\left(0\right)=-2^{n+2}(1-2^{n+2})\frac{B_{n+2}}{n+2}.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/24.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(i), §24.14 and Ch.24

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24.14.5 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)E_k\left(h\right)B_{n-k}\left(x\right)=2^n{B}_n\left(\frac{1}{2}(x+h)\right),$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $E_n\left(x\right)$: 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/24.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.14(i), §24.14 and Ch.24

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