Bernoulli and Euler polynomials

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1: 24.1 Special Notation

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Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. 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 24.3.1: Bernoulli polynomials $B_n\left(x\right)$, $n=2,3,\mathrm{\dots },6$. Magnify

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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)). …

4: 24.4 Basic Properties

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§24.4(i) Difference Equations

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§24.4(ii) Symmetry

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§24.4(iii) Sums of Powers

… ►Let $P(x)$ denote any polynomial in $x$, and after expanding set ${(B(x))}^n=B_n\left(x\right)$ and ${(E(x))}^n=E_n\left(x\right)$. … ►

24.4.39 $$E_n\left(x+h\right)={(E(x)+h)}^n.$$

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Symbols:

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

A&S Ref:

23.1.26

Permalink:

http://dlmf.nist.gov/24.4.E39

Encodings:

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See also:

Annotations for §24.4(viii), §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:

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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:

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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:

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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:

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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 $\mathrm{\ell }$ : … ►

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:

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See also:

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

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§24.16(iii) Other Generalizations

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8: 24.2 Definitions and Generating Functions

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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(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)).

10: 24.14 Sums

§24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

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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:

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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:

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See also:

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

… ►For other sums involving Bernoulli and Euler numbers and polynomials see Hansen (1975, pp. 331–347) and Prudnikov et al. (1990, pp. 383–386).