# §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 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

 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

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

## §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 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 $\displaystyle E_{n-1}\left(x\right)$ $\displaystyle=\frac{2}{n}\sum_{k=0}^{n}{n\choose k}(1-2^{k})B_{k}x^{n-k},$
 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 Ch.24

Next,

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

## §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 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

 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

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

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