§24.6 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).
Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.6

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

24.6.1 $\mathop{B_{{2n}}\/}\nolimits=\sum _{{k=2}}^{{2n+1}}\frac{(-1)^{{k-1}}}{k}{2n+1\choose k}\sum _{{j=1}}^{{k-1}}j^{{2n}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : 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

24.6.2 $\mathop{B_{{n}}\/}\nolimits=\frac{1}{n+1}\sum _{{k=1}}^{n}\sum _{{j=1}}^{k}(-1)^{j}j^{n}{\binomial{n+1}{k-j}}\Bigg/{\binomial{n}{k}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : 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

24.6.3 $\mathop{B_{{2n}}\/}\nolimits=\sum _{{k=1}}^{n}\frac{(k-1)!k!}{(2k+1)!}\*\sum _{{j=1}}^{k}(-1)^{{j-1}}{2k\choose k+j}j^{{2n}}.$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : binomial coefficient, $!$ : $n!$ : factorial, $j$ : integer, $k$ : integer and $n$ : integer
Referenced by:: §24.6
Permalink:: http://dlmf.nist.gov/24.6.E3
Encodings:: TeX, pMML, png

24.6.4 $\mathop{E_{{2n}}\/}\nolimits=\sum _{{k=1}}^{n}\frac{1}{2^{{k-1}}}\sum _{{j=1}}^{k}(-1)^{j}{2k\choose k-j}j^{{2n}},$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : 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

24.6.5 $\mathop{E_{{2n}}\/}\nolimits=\frac{1}{2^{{n-1}}}\sum _{{k=0}}^{{n-1}}(-1)^{{n-k}}(n-k)^{{2n}}\*\sum _{{j=0}}^{k}{2n-2j\choose k-j}2^{j},$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : 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

24.6.6 $\mathop{E_{{2n}}\/}\nolimits=\sum _{{k=1}}^{{2n}}\frac{(-1)^{k}}{2^{{k-1}}}{2n+1\choose k+1}\*\sum _{{j=0}}^{{\left\lfloor\tfrac{1}{2}k-\tfrac{1}{2}\right\rfloor}}{k\choose j}(k-2j)^{{2n}}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : binomial coefficient, $\left\lfloor x\right\rfloor$ : 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

24.6.7 $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)=\sum _{{k=0}}^{n}\frac{1}{k+1}\sum _{{j=0}}^{k}(-1)^{j}{k\choose j}(x+j)^{n},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : 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

24.6.8 $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)=\frac{1}{2^{n}}\sum _{{k=1}}^{{n+1}}\sum _{{j=0}}^{{k-1}}(-1)^{j}{n+1\choose k}(x+j)^{n}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : 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

24.6.9 $\mathop{B_{{n}}\/}\nolimits=\sum _{{k=0}}^{n}\frac{1}{k+1}\sum _{{j=0}}^{k}(-1)^{j}{k\choose j}j^{n},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : 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

24.6.10 $\mathop{E_{{n}}\/}\nolimits=\frac{1}{2^{n}}\sum _{{k=1}}^{{n+1}}{n+1\choose k}\sum _{{j=0}}^{{k-1}}(-1)^{j}(2j+1)^{n}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : 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

24.6.11 $\mathop{B_{{n}}\/}\nolimits=\frac{n}{2^{n}(2^{n}-1)}\sum _{{k=1}}^{n}\sum _{{j=0}}^{{k-1}}(-1)^{{j+1}}{n\choose k}j^{{n-1}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\binom{m}{n}$ : 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

24.6.12 $\mathop{E_{{2n}}\/}\nolimits=\sum _{{k=0}}^{{2n}}\frac{1}{2^{k}}\sum _{{j=0}}^{k}(-1)^{j}{k\choose j}(1+2j)^{{2n}}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\binom{m}{n}$ : 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