§24.6 Explicit Formulas

The identities in this section hold for $n=1,2,\mathrm{\dots }$. (24.6.7), (24.6.8), (24.6.10), and (24.6.12) are valid also for $n=0$.

24.6.1		$$B_{2n}=\sum _{k=2}^{2n+1}\frac{{(-1)}^{k-1}}{k}\left(\genfrac{}{}{0pt}{}{2n+1}{k}\right)\sum _{j=1}^{k-1}{j}^{2n},$$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E1 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24

24.6.2		$$B_n=\frac{1}{n+1}\sum _{k=1}^n\sum _{j=1}^k{(-1)}^j{j}^n\left(\genfrac{}{}{0.0pt}{}{n+1}{k-j}\right)/\left(\genfrac{}{}{0.0pt}{}{n}{k}\right),$$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E2 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24

24.6.3		$$B_{2n}=\sum _{k=1}^n\frac{(k-1)!k!}{(2k+1)!}\sum _{j=1}^k{(-1)}^{j-1}\left(\genfrac{}{}{0pt}{}{2k}{k+j}\right)j^{2n}.$$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E3 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24

24.6.4		$$E_{2n}=\sum _{k=1}^n\frac{1}{2^{k-1}}\sum _{j=1}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{2k}{k-j}\right)j^{2n},$$
ⓘ Symbols: $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E4 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24

24.6.5		$$E_{2n}=\frac{1}{2^{n-1}}\sum _{k=0}^{n-1}{(-1)}^{n-k}{(n-k)}^{2n}\sum _{j=0}^k\left(\genfrac{}{}{0pt}{}{2n-2j}{k-j}\right)2^j,$$
ⓘ Symbols: $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E5 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24

24.6.6		$$E_{2n}=\sum _{k=1}^{2n}\frac{{(-1)}^k}{2^{k-1}}\left(\genfrac{}{}{0pt}{}{2n+1}{k+1}\right)\sum _{j=0}^{\lfloor \frac{1}{2}k-\frac{1}{2}\rfloor }\left(\genfrac{}{}{0pt}{}{k}{j}\right){(k-2j)}^{2n}.$$
ⓘ Symbols: $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\lfloor x\rfloor$: floor of $x$, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E6 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24

24.6.7		$$B_n\left(x\right)=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(x+j)}^n,$$
ⓘ Symbols: $B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer, $n$: integer and $x$: real or complex Referenced by: §24.6, §24.6 Permalink: http://dlmf.nist.gov/24.6.E7 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24

24.6.8		$$E_n\left(x\right)=\frac{1}{2^n}\sum _{k=1}^{n+1}\sum _{j=0}^{k-1}{(-1)}^j\left(\genfrac{}{}{0pt}{}{n+1}{k}\right){(x+j)}^n.$$
ⓘ Symbols: $E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer, $n$: integer and $x$: real or complex Referenced by: §24.6, §24.6 Permalink: http://dlmf.nist.gov/24.6.E8 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24

24.6.9		$B_n$	$={\displaystyle \sum _{k=0}^n}{\displaystyle \frac{1}{k+1}}{\displaystyle \sum _{j=0}^k}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{k}{j}}\right)j^n,$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E9 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24
24.6.10		$E_n$	$={\displaystyle \frac{1}{2^n}}{\displaystyle \sum _{k=1}^{n+1}}\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{k}}\right){\displaystyle \sum _{j=0}^{k-1}}{(-1)}^j{(2j+1)}^n.$
ⓘ Symbols: $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6, §24.6 Permalink: http://dlmf.nist.gov/24.6.E10 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24

24.6.11		$$B_n=\frac{n}{2^n(2^n-1)}\sum _{k=1}^n\sum _{j=0}^{k-1}{(-1)}^{j+1}\left(\genfrac{}{}{0pt}{}{n}{k}\right)j^{n-1},$$
ⓘ Symbols: $B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6 Permalink: http://dlmf.nist.gov/24.6.E11 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24

24.6.12		$$E_{2n}=\sum _{k=0}^{2n}\frac{1}{2^k}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right){(1+2j)}^{2n}.$$
ⓘ Symbols: $E_n$: Euler numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: integer, $k$: integer and $n$: integer Referenced by: §24.6, §24.6 Permalink: http://dlmf.nist.gov/24.6.E12 Encodings: TeX, pMML, png See also: Annotations for §24.6 and Ch.24