§24.6 Explicit Formulas

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Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials, explicit formulas
Notes:: For (24.6.1) and (24.6.2) see Gould (1972, pp. 45–46). For (24.6.3) see Horata (1989). For (24.6.4) and (24.6.5) see Todorov (1978). For (24.6.6), (24.6.10), and (24.6.12) see Schwatt (1962, p. 270). For (24.6.7) see Carlitz (1961b, p. 134). For (24.6.8) and (24.6.11) see Todorov (1991, pp. 176–177). For (24.6.9) see Jordan (1965, p. 236).
Permalink:: http://dlmf.nist.gov/24.6
See also:: Annotations for Ch.24

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

24.6.1		$B_{2n}=\sum_{k=2}^{2n+1}\frac{(-1)^{k-1}}{k}{2n+1\choose k}\sum_{j=1}^{k-1}j^{% 2n},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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		$B_{n}=\frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{n}{\genfrac{(}{)}{0.% 0pt}{}{n+1}{k-j}}\Bigg{/}{\genfrac{(}{)}{0.0pt}{}{n}{k}},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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		$B_{2n}=\sum_{k=1}^{n}\frac{(k-1)!k!}{(2k+1)!}\*\sum_{j=1}^{k}(-1)^{j-1}{2k% \choose k+j}j^{2n}.$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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		$E_{2n}=\sum_{k=1}^{n}\frac{1}{2^{k-1}}\sum_{j=1}^{k}(-1)^{j}{2k\choose k-j}j^{% 2n},$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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		$E_{2n}=\frac{1}{2^{n-1}}\sum_{k=0}^{n-1}(-1)^{n-k}(n-k)^{2n}\*\sum_{j=0}^{k}{2% n-2j\choose k-j}2^{j},$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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		$E_{2n}=\sum_{k=1}^{2n}\frac{(-1)^{k}}{2^{k-1}}{2n+1\choose k+1}\*\sum_{j=0}^{% \left\lfloor\tfrac{1}{2}k-\tfrac{1}{2}\right\rfloor}{k\choose j}(k-2j)^{2n}.$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\left\lfloor\NVar{x}\right\rfloor$ : 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		$B_{n}\left(x\right)=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j% }(x+j)^{n},$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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		$E_{n}\left(x\right)=\frac{1}{2^{n}}\sum_{k=1}^{n+1}\sum_{j=0}^{k-1}(-1)^{j}{n+% 1\choose k}(x+j)^{n}.$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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	$\displaystyle B_{n}$	$\displaystyle=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j}j^{% n},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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	$\displaystyle E_{n}$	$\displaystyle=\frac{1}{2^{n}}\sum_{k=1}^{n+1}{n+1\choose k}\sum_{j=0}^{k-1}(-1% )^{j}(2j+1)^{n}.$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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		$B_{n}=\frac{n}{2^{n}(2^{n}-1)}\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j+1}{n% \choose k}j^{n-1},$
ⓘ Symbols: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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		$E_{2n}=\sum_{k=0}^{2n}\frac{1}{2^{k}}\sum_{j=0}^{k}(-1)^{j}{k\choose j}(1+2j)^% {2n}.$
ⓘ Symbols: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : 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