24 Bernoulli and Euler Polynomials Properties 24.3 Graphs 24.5 Recurrence Relations

§24.4 Basic Properties

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Permalink:: http://dlmf.nist.gov/24.4
See also:: Annotations for Ch.24

§24.4(i) Difference Equations

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Keywords:: Bernoulli polynomials, Euler polynomials, difference equation
Notes:: See Nörlund (1924, pp. 18 and 23) or Milne-Thomson (1933, pp. 136 and 146).
Permalink:: http://dlmf.nist.gov/24.4.i
See also:: Annotations for §24.4 and Ch.24

24.4.1	$\displaystyle B_{n}\left(x+1\right)-B_{n}\left(x\right)$	$\displaystyle=nx^{n-1},$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.6 Referenced by: (13.8.16), §24.13(i) Permalink: http://dlmf.nist.gov/24.4.E1 Encodings: TeX, pMML, png See also: Annotations for §24.4(i), §24.4 and Ch.24
24.4.2	$\displaystyle E_{n}\left(x+1\right)+E_{n}\left(x\right)$	$\displaystyle=2x^{n}.$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.6 Permalink: http://dlmf.nist.gov/24.4.E2 Encodings: TeX, pMML, png See also: Annotations for §24.4(i), §24.4 and Ch.24

§24.4(ii) Symmetry

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Keywords:: Bernoulli polynomials, Euler polynomials, symmetry
Notes:: See Nörlund (1924, pp. 21 and 24) or Milne-Thomson (1933, pp. 136 and 146).
Referenced by:: §24.4(vi)
Permalink:: http://dlmf.nist.gov/24.4.ii
See also:: Annotations for §24.4 and Ch.24

24.4.3	$\displaystyle B_{n}\left(1-x\right)$	$\displaystyle=(-1)^{n}B_{n}\left(x\right),$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.8 Permalink: http://dlmf.nist.gov/24.4.E3 Encodings: TeX, pMML, png See also: Annotations for §24.4(ii), §24.4 and Ch.24
24.4.4	$\displaystyle E_{n}\left(1-x\right)$	$\displaystyle=(-1)^{n}E_{n}\left(x\right).$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.8 Permalink: http://dlmf.nist.gov/24.4.E4 Encodings: TeX, pMML, png See also: Annotations for §24.4(ii), §24.4 and Ch.24
24.4.5	$\displaystyle(-1)^{n}B_{n}\left(-x\right)$	$\displaystyle=B_{n}\left(x\right)+nx^{n-1},$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.9 Permalink: http://dlmf.nist.gov/24.4.E5 Encodings: TeX, pMML, png See also: Annotations for §24.4(ii), §24.4 and Ch.24
24.4.6	$\displaystyle(-1)^{n+1}E_{n}\left(-x\right)$	$\displaystyle=E_{n}\left(x\right)-2x^{n}.$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.9 Permalink: http://dlmf.nist.gov/24.4.E6 Encodings: TeX, pMML, png See also: Annotations for §24.4(ii), §24.4 and Ch.24

§24.4(iii) Sums of Powers

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Keywords:: Bernoulli polynomials, Euler polynomials, as Bernoulli or Euler polynomials, representation as sums of powers, representations as sums of powers, sums of powers
Notes:: See Nörlund (1924, p. 19) or Milne-Thomson (1933, pp. 137 and 147). For (24.4.9) and (24.4.10) see Howard (1996b) and Slavutskiĭ (2000). For (24.4.11) see Apostol (2006).
Permalink:: http://dlmf.nist.gov/24.4.iii
See also:: Annotations for §24.4 and Ch.24

24.4.7	$\displaystyle\sum_{k=1}^{m}k^{n}$	$\displaystyle=\frac{B_{n+1}\left(m+1\right)-B_{n+1}}{n+1},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $k$ : integer, $m$ : integer and $n$ : integer A&S Ref: 23.1.4 Referenced by: §24.17(iii) Permalink: http://dlmf.nist.gov/24.4.E7 Encodings: TeX, pMML, png See also: Annotations for §24.4(iii), §24.4 and Ch.24
24.4.8	$\displaystyle\sum_{k=1}^{m}(-1)^{m-k}k^{n}$	$\displaystyle=\frac{E_{n}\left(m+1\right)+(-1)^{m}E_{n}\left(0\right)}{2}.$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $k$ : integer, $m$ : integer and $n$ : integer A&S Ref: 23.1.4 Referenced by: §24.17(iii) Permalink: http://dlmf.nist.gov/24.4.E8 Encodings: TeX, pMML, png See also: Annotations for §24.4(iii), §24.4 and Ch.24

24.4.9	$\displaystyle\sum_{k=0}^{m-1}(a+dk)^{n}$	$\displaystyle={\frac{d^{n}}{n+1}\left(B_{n+1}\left(m+\frac{a}{d}\right)-B_{n+1% }\left(\frac{a}{d}\right)\right)},$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $k$ : integer, $m$ : integer and $n$ : integer Referenced by: §24.4(iii) Permalink: http://dlmf.nist.gov/24.4.E9 Encodings: TeX, pMML, png See also: Annotations for §24.4(iii), §24.4 and Ch.24
24.4.10	$\displaystyle\sum_{k=0}^{m-1}(-1)^{k}(a+dk)^{n}$	$\displaystyle={\frac{d^{n}}{2}\left((-1)^{m-1}E_{n}\left(m+\frac{a}{d}\right)+% E_{n}\left(\frac{a}{d}\right)\right)}.$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $k$ : integer, $m$ : integer and $n$ : integer Referenced by: §24.4(iii) Permalink: http://dlmf.nist.gov/24.4.E10 Encodings: TeX, pMML, png See also: Annotations for §24.4(iii), §24.4 and Ch.24
24.4.11	$\displaystyle\sum_{\begin{subarray}{c}k=1\\ \left(k,m\right)=1\end{subarray}}^{m}k^{n}$	$\displaystyle=\frac{1}{n+1}\sum_{j=1}^{n+1}{n+1\choose j}\*\left(\prod_{p% \mathbin{\|}m}(1-p^{n-j})B_{n+1-j}\right)m^{j}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left(\NVar{m},\NVar{n}\right)$ : greatest common divisor (gcd), $j$ : integer, $k$ : integer, $m$ : integer, $n$ : integer and $p$ : prime Referenced by: §24.4(iii) Permalink: http://dlmf.nist.gov/24.4.E11 Encodings: TeX, pMML, png See also: Annotations for §24.4(iii), §24.4 and Ch.24

§24.4(iv) Finite Expansions

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Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials, finite expansions
Notes:: These identities follow from (24.2.3) and (24.2.8).
Permalink:: http://dlmf.nist.gov/24.4.iv
See also:: Annotations for §24.4 and Ch.24

24.4.12	$\displaystyle B_{n}\left(x+h\right)$	$\displaystyle=\sum_{k=0}^{n}{n\choose k}B_{k}\left(x\right)h^{n-k},$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.7 Permalink: http://dlmf.nist.gov/24.4.E12 Encodings: TeX, pMML, png See also: Annotations for §24.4(iv), §24.4 and Ch.24
24.4.13	$\displaystyle E_{n}\left(x+h\right)$	$\displaystyle=\sum_{k=0}^{n}{n\choose k}E_{k}\left(x\right)h^{n-k},$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.7 Permalink: http://dlmf.nist.gov/24.4.E13 Encodings: TeX, pMML, png See also: Annotations for §24.4(iv), §24.4 and Ch.24
24.4.14	$\displaystyle E_{n-1}\left(x\right)$	$\displaystyle=\frac{2}{n}\sum_{k=0}^{n}{n\choose k}(1-2^{k})B_{k}x^{n-k},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer, $n$ : integer and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.4.E14 Encodings: TeX, pMML, png See also: Annotations for §24.4(iv), §24.4 and Ch.24

24.4.15	$\displaystyle B_{2n}$	$\displaystyle=\frac{2n}{2^{2n}(2^{2n}-1)}\sum_{k=0}^{n-1}{2n-1\choose 2k}E_{2k},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer Permalink: http://dlmf.nist.gov/24.4.E15 Encodings: TeX, pMML, png See also: Annotations for §24.4(iv), §24.4 and Ch.24
24.4.16	$\displaystyle E_{2n}$	$\displaystyle=\frac{1}{2n+1}-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k-% 1}-1)B_{2k}}{k},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer Permalink: http://dlmf.nist.gov/24.4.E16 Encodings: TeX, pMML, png See also: Annotations for §24.4(iv), §24.4 and Ch.24
24.4.17	$\displaystyle E_{2n}$	$\displaystyle=1-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k}-1)B_{2k}}{2k}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $k$ : integer and $n$ : integer Permalink: http://dlmf.nist.gov/24.4.E17 Encodings: TeX, pMML, png See also: Annotations for §24.4(iv), §24.4 and Ch.24

§24.4(v) Multiplication Formulas

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Keywords:: Bernoulli polynomials, multiplication formulas
Notes:: See Nörlund (1924, pp. 21 and 24) or Milne-Thomson (1933, p. 141). For (24.4.24) see Todorov (1991).
Referenced by:: §24.4(vi)
Permalink:: http://dlmf.nist.gov/24.4.v
See also:: Annotations for §24.4 and Ch.24

Raabe’s Theorem

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Keywords:: Bernoulli polynomials, Euler polynomials, Raabe’s theorem, multiplication formulas
See also:: Annotations for §24.4(v), §24.4 and Ch.24

24.4.18		$B_{n}\left(mx\right)=m^{n-1}\sum_{k=0}^{m-1}B_{n}\left(x+\frac{k}{m}\right).$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $k$ : integer, $m$ : integer, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.10 Permalink: http://dlmf.nist.gov/24.4.E18 Encodings: TeX, pMML, png See also: Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24

Next,

24.4.19		$E_{n}\left(mx\right)=-\frac{2m^{n}}{n+1}\sum_{k=0}^{m-1}(-1)^{k}B_{n+1}\left(x% +\frac{k}{m}\right),$
$m=2,4,6,\dots$ ,
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $k$ : integer, $m$ : integer, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.10 Permalink: http://dlmf.nist.gov/24.4.E19 Encodings: TeX, pMML, png See also: Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24

24.4.20		$E_{n}\left(mx\right)=m^{n}\sum_{k=0}^{m-1}(-1)^{k}E_{n}\left(x+\frac{k}{m}% \right),$
$m=1,3,5,\dots$ .
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $k$ : integer, $m$ : integer, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.10 Permalink: http://dlmf.nist.gov/24.4.E20 Encodings: TeX, pMML, png See also: Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24

24.4.21	$\displaystyle B_{n}\left(x\right)$	$\displaystyle=2^{n-1}\left(B_{n}\left(\tfrac{1}{2}x\right)+B_{n}\left(\tfrac{1% }{2}x+\tfrac{1}{2}\right)\right),$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex Permalink: http://dlmf.nist.gov/24.4.E21 Encodings: TeX, pMML, png See also: Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24
24.4.22	$\displaystyle E_{n-1}\left(x\right)$	$\displaystyle=\frac{2}{n}\left(B_{n}\left(x\right)-2^{n}B_{n}\left(\tfrac{1}{2% }x\right)\right),$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.27 Permalink: http://dlmf.nist.gov/24.4.E22 Encodings: TeX, pMML, png See also: Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24
24.4.23	$\displaystyle E_{n-1}\left(x\right)$	$\displaystyle=\frac{2^{n}}{n}\left(B_{n}\left(\tfrac{1}{2}x+\tfrac{1}{2}\right% )-B_{n}\left(\tfrac{1}{2}x\right)\right),$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.27 Permalink: http://dlmf.nist.gov/24.4.E23 Encodings: TeX, pMML, png See also: Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24

24.4.24		$B_{n}\left(mx\right)=m^{n}B_{n}\left(x\right)+n\sum_{k=1}^{n}\sum_{j=0}^{k-1}(% -1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}\frac{e^{2\pi i(k-j)r/m}}{(1-e^{2% \pi ir/m})^{n}}\right)(j+mx)^{n-1},$
$n=1,2,\dots$ , $m=2,3,\dots$ .
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $j$ : integer, $k$ : integer, $m$ : integer, $n$ : integer and $x$ : real or complex Referenced by: §24.4(v) Permalink: http://dlmf.nist.gov/24.4.E24 Encodings: TeX, pMML, png See also: Annotations for §24.4(v), §24.4(v), §24.4 and Ch.24

§24.4(vi) Special Values

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Keywords:: Bernoulli polynomials, Euler polynomials, special values
Notes:: (24.4.25)–(24.4.33) follow from the identities in §§24.4(ii) and 24.4(v).
Permalink:: http://dlmf.nist.gov/24.4.vi
See also:: Annotations for §24.4 and Ch.24

24.4.25		$B_{n}\left(0\right)=(-1)^{n}B_{n}\left(1\right)=B_{n}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer A&S Ref: 23.1.20 Referenced by: §24.17(ii), §24.4(vi) Permalink: http://dlmf.nist.gov/24.4.E25 Encodings: TeX, pMML, png See also: Annotations for §24.4(vi), §24.4 and Ch.24

24.4.26		$E_{n}\left(0\right)=-E_{n}\left(1\right)=-\frac{2}{n+1}(2^{n+1}-1)B_{n+1},$
$n>0$ .
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : 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: TeX, pMML, png 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

24.4.27	$\displaystyle B_{n}\left(\tfrac{1}{2}\right)$	$\displaystyle=-(1-2^{1-n})B_{n},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer A&S Ref: 23.1.21 Referenced by: §2.10(i), §24.17(ii) Permalink: http://dlmf.nist.gov/24.4.E27 Encodings: TeX, pMML, png See also: Annotations for §24.4(vi), §24.4 and Ch.24
24.4.28	$\displaystyle E_{n}\left(\tfrac{1}{2}\right)$	$\displaystyle=2^{-n}E_{n}.$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials and $n$ : integer A&S Ref: 23.1.21 Referenced by: §24.9 Permalink: http://dlmf.nist.gov/24.4.E28 Encodings: TeX, pMML, png See also: Annotations for §24.4(vi), §24.4 and Ch.24

24.4.29		$B_{2n}\left(\tfrac{1}{3}\right)=B_{2n}\left(\tfrac{2}{3}\right)=-\tfrac{1}{2}(% 1-3^{1-2n})B_{2n}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer A&S Ref: 23.1.23 Permalink: http://dlmf.nist.gov/24.4.E29 Encodings: TeX, pMML, png See also: Annotations for §24.4(vi), §24.4 and Ch.24

24.4.30		$E_{2n-1}\left(\tfrac{1}{3}\right)=-E_{2n-1}\left(\tfrac{2}{3}\right)=-\frac{(1% -3^{1-2n})(2^{2n}-1)}{2n}B_{2n},$
$n=1,2,\dots$ .
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials and $n$ : integer A&S Ref: 23.1.22 Permalink: http://dlmf.nist.gov/24.4.E30 Encodings: TeX, pMML, png See also: Annotations for §24.4(vi), §24.4 and Ch.24

24.4.31		$B_{n}\left(\tfrac{1}{4}\right)=(-1)^{n}B_{n}\left(\tfrac{3}{4}\right)=-\frac{1% -2^{1-n}}{2^{n}}B_{n}-\frac{n}{4^{n}}E_{n-1},$
$n=1,2,\dots$ .
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $E_{\NVar{n}}$ : Euler numbers and $n$ : integer A&S Ref: 23.1.22 Permalink: http://dlmf.nist.gov/24.4.E31 Encodings: TeX, pMML, png See also: Annotations for §24.4(vi), §24.4 and Ch.24

24.4.32		$B_{2n}\left(\tfrac{1}{6}\right)=B_{2n}\left(\tfrac{5}{6}\right)=\tfrac{1}{2}(1% -2^{1-2n})(1-3^{1-2n})B_{2n},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials and $n$ : integer A&S Ref: 23.1.24 Permalink: http://dlmf.nist.gov/24.4.E32 Encodings: TeX, pMML, png See also: Annotations for §24.4(vi), §24.4 and Ch.24

24.4.33		$E_{2n}\left(\tfrac{1}{6}\right)=E_{2n}\left(\tfrac{5}{6}\right)=\frac{1+3^{-2n% }}{2^{2n+1}}E_{2n}.$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials and $n$ : integer Referenced by: §24.4(vi) Permalink: http://dlmf.nist.gov/24.4.E33 Encodings: TeX, pMML, png See also: Annotations for §24.4(vi), §24.4 and Ch.24

§24.4(vii) Derivatives

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Keywords:: Bernoulli polynomials, Euler polynomials, derivative
Notes:: Use (24.2.3) and (24.2.8).
Permalink:: http://dlmf.nist.gov/24.4.vii
See also:: Annotations for §24.4 and Ch.24

24.4.34	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}B_{n}\left(x\right)$	$\displaystyle=nB_{n-1}\left(x\right),$
$n=1,2,\dots$ ,
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer and $x$ : real or complex A&S Ref: 23.1.5 Referenced by: §24.13(i), (25.11.7) Permalink: http://dlmf.nist.gov/24.4.E34 Encodings: TeX, pMML, png See also: Annotations for §24.4(vii), §24.4 and Ch.24
24.4.35	$\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}E_{n}\left(x\right)$	$\displaystyle=nE_{n-1}\left(x\right),$
$n=1,2,\dots$ .
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $n$ : integer and $x$ : real or complex A&S Ref: 23.1.5 Referenced by: §24.9 Permalink: http://dlmf.nist.gov/24.4.E35 Encodings: TeX, pMML, png See also: Annotations for §24.4(vii), §24.4 and Ch.24

§24.4(viii) Symbolic Operations

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Keywords:: Bernoulli and Euler polynomials, Bernoulli polynomials, Euler polynomials, symbolic operations, umbral calculus
Permalink:: http://dlmf.nist.gov/24.4.viii
See also:: Annotations for §24.4 and Ch.24

Let $P(x)$ denote any polynomial in $x$ , and after expanding set $(B(x))^{n}=B_{n}\left(x\right)$ and $(E(x))^{n}=E_{n}\left(x\right)$ . Then

24.4.36	$\displaystyle P(B(x)+1)-P(B(x))$	$\displaystyle=P^{\prime}(x),$
ⓘ Symbols: $x$ : real or complex A&S Ref: 23.1.25 Permalink: http://dlmf.nist.gov/24.4.E36 Encodings: TeX, pMML, png See also: Annotations for §24.4(viii), §24.4 and Ch.24
24.4.37	$\displaystyle B_{n}\left(x+h\right)$	$\displaystyle=(B(x)+h)^{n},$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.26 Permalink: http://dlmf.nist.gov/24.4.E37 Encodings: TeX, pMML, png See also: Annotations for §24.4(viii), §24.4 and Ch.24
24.4.38	$\displaystyle P(E(x)+1)+P(E(x))$	$\displaystyle=2P(x),$
ⓘ Symbols: $x$ : real or complex A&S Ref: 23.1.25 Permalink: http://dlmf.nist.gov/24.4.E38 Encodings: TeX, pMML, png See also: Annotations for §24.4(viii), §24.4 and Ch.24
24.4.39	$\displaystyle E_{n}\left(x+h\right)$	$\displaystyle=(E(x)+h)^{n}.$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $n$ : integer and $x$ : real or complex A&S Ref: 23.1.26 Permalink: http://dlmf.nist.gov/24.4.E39 Encodings: TeX, pMML, png See also: Annotations for §24.4(viii), §24.4 and Ch.24

For these results and also connections with the umbral calculus see Gessel (2003).

§24.4(ix) Relations to Other Functions

ⓘ

Keywords:: Bernoulli numbers, Bernoulli polynomials, Eulerian numbers, relation to Bernoulli numbers, relation to Eulerian numbers, relation to Riemann zeta function, relations to
Permalink:: http://dlmf.nist.gov/24.4.ix
See also:: Annotations for §24.4 and Ch.24

For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).

24.3 Graphs 24.5 Recurrence Relations

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