§24.4 Basic Properties

§24.4(i) Difference Equations

 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 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 24

§24.4(ii) Symmetry

 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 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 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 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 24

§24.4(iii) Sums of Powers

 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 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 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 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 24 24.4.11 $\displaystyle\sum_{\begin{subarray}{c}k=1\\ \left(k,m\right)=1\end{subarray}}^{m}k^{n}$ $\displaystyle=\frac{1}{n+1}\sum_{j=1}^{n+1}{n+1\choose j}\*\left(\prod_{p% \mathbin{|}m}(1-p^{n-j})B_{n+1-j}\right)m^{j}.$

§24.4(iv) Finite Expansions

 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 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 24 24.4.14 $\displaystyle E_{n-1}\left(x\right)$ $\displaystyle=\frac{2}{n}\sum_{k=0}^{n}{n\choose k}(1-2^{k})B_{k}x^{n-k},$
 24.4.15 $\displaystyle B_{2n}$ $\displaystyle=\frac{2n}{2^{2n}(2^{2n}-1)}\sum_{k=0}^{n-1}{2n-1\choose 2k}E_{2k},$ 24.4.16 $\displaystyle E_{2n}$ $\displaystyle=\frac{1}{2n+1}-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k-% 1}-1)B_{2k}}{k},$ 24.4.17 $\displaystyle E_{2n}$ $\displaystyle=1-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k}-1)B_{2k}}{2k}.$

§24.4(v) Multiplication Formulas

Raabe’s Theorem

 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 24

Next,

 24.4.19 $E_{n}\left(mx\right)=-\frac{2m^{n}}{n+1}\sum_{k=0}^{m-1}(-1)^{k}B_{n+1}\left(x% +\frac{k}{m}\right),$ $m=2,4,6,\dots$,
 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 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 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 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 24
 24.4.24 $B_{n}\left(mx\right)=m^{n}B_{n}\left(x\right)+n\sum_{k=1}^{n}\sum_{j=0}^{k-1}(% -1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}\frac{e^{2\pi i(k-j)r/m}}{(1-e^{2% \pi ir/m})^{n}}\right)(j+mx)^{n-1},$ $n=1,2,\dots$, $m=2,3,\dots$.

§24.4(vi) Special Values

 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 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, 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 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 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 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 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 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 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 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 24

§24.4(vii) Derivatives

 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) Permalink: http://dlmf.nist.gov/24.4.E34 Encodings: TeX, pMML, png See also: Annotations for 24.4(vii), 24.4 and 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 24

§24.4(viii) Symbolic Operations

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 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 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 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 24

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

§24.4(ix) Relations to Other Functions

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