24 Bernoulli and Euler Polynomials Properties 24.5 Recurrence Relations 24.7 Integral Representations

§24.6 Explicit Formulas

ⓘ

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 .

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, : integer, : integer and : 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, : integer, : integer and : 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 ), : integer, : integer and : 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, : integer, : integer and : 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, : integer, : integer and : 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 , : integer, : integer and : 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, : integer, : integer, : integer and : 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, : integer, : integer, : integer and : 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 $B_{n}=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j}j^{n},$

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : integer, : integer and : 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 $E_{n}=\frac{1}{2^{n}}\sum_{k=1}^{n+1}{n+1\choose k}\sum_{j=0}^{k-1}(-1)^{j}(2j% +1)^{n}.$

ⓘ

Symbols:: $E_{\NVar{n}}$ : Euler numbers, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, : integer, : integer and : 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, : integer, : integer and : 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, : integer, : integer and : 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

24.5 Recurrence Relations 24.7 Integral Representations

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov