Bernoulli and Euler Polynomials - 24.6 Explicit Formulas

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
24.6.E1 B 2 ⁢ n = ∑ k = 2 2 ⁢ n + 1 ( - 1 ) k - 1 k ⁢ ( 2 ⁢ n + 1 k ) ⁢ ∑ j = 1 k - 1 j 2 ⁢ n Bernoulli-number-B 2 𝑛 superscript subscript 𝑘 2 2 𝑛 1 superscript 1 𝑘 1 𝑘 binomial 2 𝑛 1 𝑘 superscript subscript 𝑗 1 𝑘 1 superscript 𝑗 2 𝑛 {\displaystyle{\displaystyle B_{2n}=\sum_{k=2}^{2n+1}\frac{(-1)^{k-1}}{k}{2n+1% \choose k}\sum_{j=1}^{k-1}j^{2n}}}
\BernoullinumberB{2n} = \sum_{k=2}^{2n+1}\frac{(-1)^{k-1}}{k}{2n+1\choose k}\sum_{j=1}^{k-1}j^{2n}		 

bernoulli(2*n) = sum(((- 1)^(k - 1))/(k)*binomial(2*n + 1,k)*sum((j)^(2*n), j = 1..k - 1), k = 2..2*n + 1)

BernoulliB[2*n] == Sum[Divide[(- 1)^(k - 1),k]*Binomial[2*n + 1,k]*Sum[(j)^(2*n), {j, 1, k - 1}, GenerateConditions->None], {k, 2, 2*n + 1}, GenerateConditions->None]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.6.E2 B n = 1 n + 1 ⁢ ∑ k = 1 n ∑ j = 1 k ( - 1 ) j ⁢ j n ⁢ ( n + 1 k - j ) / ( n k ) Bernoulli-number-B 𝑛 1 𝑛 1 superscript subscript 𝑘 1 𝑛 superscript subscript 𝑗 1 𝑘 superscript 1 𝑗 superscript 𝑗 𝑛 binomial 𝑛 1 𝑘 𝑗 binomial 𝑛 𝑘 {\displaystyle{\displaystyle 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}}}}
\BernoullinumberB{n} = \frac{1}{n+1}\sum_{k=1}^{n}\sum_{j=1}^{k}(-1)^{j}j^{n}{\binom{n+1}{k-j}}\Bigg{/}{\binom{n}{k}}		 

bernoulli(n) = (1)/(n + 1)*sum(sum((- 1)^(j)* (j)^(n)*binomial(n + 1,k - j)/(binomial(n,k)), j = 1..k), k = 1..n)

BernoulliB[n] == Divide[1,n + 1]*Sum[Sum[(- 1)^(j)* (j)^(n)*Binomial[n + 1,k - j]/(Binomial[n,k]), {j, 1, k}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]
Aborted Failure Successful [Tested: 3] Successful [Tested: 3]
24.6.E3 B 2 ⁢ n = ∑ k = 1 n ( k - 1 ) ! ⁢ k ! ( 2 ⁢ k + 1 ) ! ⁢ ∑ j = 1 k ( - 1 ) j - 1 ⁢ ( 2 ⁢ k k + j ) ⁢ j 2 ⁢ n Bernoulli-number-B 2 𝑛 superscript subscript 𝑘 1 𝑛 𝑘 1 𝑘 2 𝑘 1 superscript subscript 𝑗 1 𝑘 superscript 1 𝑗 1 binomial 2 𝑘 𝑘 𝑗 superscript 𝑗 2 𝑛 {\displaystyle{\displaystyle 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}}}
\BernoullinumberB{2n} = \sum_{k=1}^{n}\frac{(k-1)!k!}{(2k+1)!}\*\sum_{j=1}^{k}(-1)^{j-1}{2k\choose k+j}j^{2n}		 

bernoulli(2*n) = sum((factorial(k - 1)*factorial(k))/(factorial(2*k + 1))* sum((- 1)^(j - 1)*binomial(2*k,k + j)*(j)^(2*n), j = 1..k), k = 1..n)

BernoulliB[2*n] == Sum[Divide[(k - 1)!*(k)!,(2*k + 1)!]* Sum[(- 1)^(j - 1)*Binomial[2*k,k + j]*(j)^(2*n), {j, 1, k}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]
Aborted Failure
Failed [3 / 3]
Result: .1666666667
Test Values: {n = 1}


Result: -.3333333333e-1
Test Values: {n = 2}

... skip entries to safe data
 Successful [Tested: 3]
24.6.E4 E 2 ⁢ n = ∑ k = 1 n 1 2 k - 1 ⁢ ∑ j = 1 k ( - 1 ) j ⁢ ( 2 ⁢ k k - j ) ⁢ j 2 ⁢ n Euler-number-E 2 𝑛 superscript subscript 𝑘 1 𝑛 1 superscript 2 𝑘 1 superscript subscript 𝑗 1 𝑘 superscript 1 𝑗 binomial 2 𝑘 𝑘 𝑗 superscript 𝑗 2 𝑛 {\displaystyle{\displaystyle E_{2n}=\sum_{k=1}^{n}\frac{1}{2^{k-1}}\sum_{j=1}^% {k}(-1)^{j}{2k\choose k-j}j^{2n}}}
\EulernumberE{2n} = \sum_{k=1}^{n}\frac{1}{2^{k-1}}\sum_{j=1}^{k}(-1)^{j}{2k\choose k-j}j^{2n}		 

euler(2*n) = sum((1)/((2)^(k - 1))*sum((- 1)^(j)*binomial(2*k,k - j)*(j)^(2*n), j = 1..k), k = 1..n)

EulerE[2*n] == Sum[Divide[1,(2)^(k - 1)]*Sum[(- 1)^(j)*Binomial[2*k,k - j]*(j)^(2*n), {j, 1, k}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]
Missing Macro Error Failure - Successful [Tested: 3]
24.6.E5 E 2 ⁢ n = 1 2 n - 1 ⁢ ∑ k = 0 n - 1 ( - 1 ) n - k ⁢ ( n - k ) 2 ⁢ n ⁢ ∑ j = 0 k ( 2 ⁢ n - 2 ⁢ j k - j ) ⁢ 2 j Euler-number-E 2 𝑛 1 superscript 2 𝑛 1 superscript subscript 𝑘 0 𝑛 1 superscript 1 𝑛 𝑘 superscript 𝑛 𝑘 2 𝑛 superscript subscript 𝑗 0 𝑘 binomial 2 𝑛 2 𝑗 𝑘 𝑗 superscript 2 𝑗 {\displaystyle{\displaystyle E_{2n}=\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}}}
\EulernumberE{2n} = \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}		 

euler(2*n) = (1)/((2)^(n - 1))*sum((- 1)^(n - k)*(n - k)^(2*n)* sum(binomial(2*n - 2*j,k - j)*(2)^(j), j = 0..k), k = 0..n - 1)

EulerE[2*n] == Divide[1,(2)^(n - 1)]*Sum[(- 1)^(n - k)*(n - k)^(2*n)* Sum[Binomial[2*n - 2*j,k - j]*(2)^(j), {j, 0, k}, GenerateConditions->None], {k, 0, n - 1}, GenerateConditions->None]
Missing Macro Error Failure - Successful [Tested: 3]
24.6.E6 E 2 ⁢ n = ∑ k = 1 2 ⁢ n ( - 1 ) k 2 k - 1 ⁢ ( 2 ⁢ n + 1 k + 1 ) ⁢ ∑ j = 0 ⌊ 1 2 ⁢ k - 1 2 ⌋ ( k j ) ⁢ ( k - 2 ⁢ j ) 2 ⁢ n Euler-number-E 2 𝑛 superscript subscript 𝑘 1 2 𝑛 superscript 1 𝑘 superscript 2 𝑘 1 binomial 2 𝑛 1 𝑘 1 superscript subscript 𝑗 0 1 2 𝑘 1 2 binomial 𝑘 𝑗 superscript 𝑘 2 𝑗 2 𝑛 {\displaystyle{\displaystyle 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}}}
\EulernumberE{2n} = \sum_{k=1}^{2n}\frac{(-1)^{k}}{2^{k-1}}{2n+1\choose k+1}\*\sum_{j=0}^{\floor{\tfrac{1}{2}k-\tfrac{1}{2}}}{k\choose j}(k-2j)^{2n}		 

euler(2*n) = sum(((- 1)^(k))/((2)^(k - 1))*binomial(2*n + 1,k + 1)* sum(binomial(k,j)*(k - 2*j)^(2*n), j = 0..floor((1)/(2)*k -(1)/(2))), k = 1..2*n)

EulerE[2*n] == Sum[Divide[(- 1)^(k),(2)^(k - 1)]*Binomial[2*n + 1,k + 1]* Sum[Binomial[k,j]*(k - 2*j)^(2*n), {j, 0, Floor[Divide[1,2]*k -Divide[1,2]]}, GenerateConditions->None], {k, 1, 2*n}, GenerateConditions->None]
Missing Macro Error Failure - Successful [Tested: 3]
24.6.E7 B n ⁡ ( x ) = ∑ k = 0 n 1 k + 1 ⁢ ∑ j = 0 k ( - 1 ) j ⁢ ( k j ) ⁢ ( x + j ) n Bernoulli-polynomial-B 𝑛 𝑥 superscript subscript 𝑘 0 𝑛 1 𝑘 1 superscript subscript 𝑗 0 𝑘 superscript 1 𝑗 binomial 𝑘 𝑗 superscript 𝑥 𝑗 𝑛 {\displaystyle{\displaystyle 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}}}
\BernoullipolyB{n}@{x} = \sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j}(x+j)^{n}		 

bernoulli(n, x) = sum((1)/(k + 1)*sum((- 1)^(j)*binomial(k,j)*(x + j)^(n), j = 0..k), k = 0..n)

BernoulliB[n, x] == Sum[Divide[1,k + 1]*Sum[(- 1)^(j)*Binomial[k,j]*(x + j)^(n), {j, 0, k}, GenerateConditions->None], {k, 0, n}, GenerateConditions->None]
Failure Failure
Failed [7 / 9]
Result: 1.
Test Values: {x = 3/2, n = 1}


Result: .9166666667
Test Values: {x = 3/2, n = 2}

... skip entries to safe data
 Successful [Tested: 9]
24.6.E8 E n ⁡ ( x ) = 1 2 n ⁢ ∑ k = 1 n + 1 ∑ j = 0 k - 1 ( - 1 ) j ⁢ ( n + 1 k ) ⁢ ( x + j ) n Euler-polynomial-E 𝑛 𝑥 1 superscript 2 𝑛 superscript subscript 𝑘 1 𝑛 1 superscript subscript 𝑗 0 𝑘 1 superscript 1 𝑗 binomial 𝑛 1 𝑘 superscript 𝑥 𝑗 𝑛 {\displaystyle{\displaystyle 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}}}
\EulerpolyE{n}@{x} = \frac{1}{2^{n}}\sum_{k=1}^{n+1}\sum_{j=0}^{k-1}(-1)^{j}{n+1\choose k}(x+j)^{n}		 

euler(n, x) = (1)/((2)^(n))*sum(sum((- 1)^(j)*binomial(n + 1,k)*(x + j)^(n), j = 0..k - 1), k = 1..n + 1)

EulerE[n, x] == Divide[1,(2)^(n)]*Sum[Sum[(- 1)^(j)*Binomial[n + 1,k]*(x + j)^(n), {j, 0, k - 1}, GenerateConditions->None], {k, 1, n + 1}, GenerateConditions->None]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
24.6.E9 B n = ∑ k = 0 n 1 k + 1 ⁢ ∑ j = 0 k ( - 1 ) j ⁢ ( k j ) ⁢ j n Bernoulli-number-B 𝑛 superscript subscript 𝑘 0 𝑛 1 𝑘 1 superscript subscript 𝑗 0 𝑘 superscript 1 𝑗 binomial 𝑘 𝑗 superscript 𝑗 𝑛 {\displaystyle{\displaystyle B_{n}=\sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-% 1)^{j}{k\choose j}j^{n}}}
\BernoullinumberB{n} = \sum_{k=0}^{n}\frac{1}{k+1}\sum_{j=0}^{k}(-1)^{j}{k\choose j}j^{n}		 

bernoulli(n) = sum((1)/(k + 1)*sum((- 1)^(j)*binomial(k,j)*(j)^(n), j = 0..k), k = 0..n)

BernoulliB[n] == Sum[Divide[1,k + 1]*Sum[(- 1)^(j)*Binomial[k,j]*(j)^(n), {j, 0, k}, GenerateConditions->None], {k, 0, n}, GenerateConditions->None]
Failure Successful
Failed [2 / 3]
Result: -.5000000000
Test Values: {n = 1}


Result: .1666666667
Test Values: {n = 2}

Successful [Tested: 3]
24.6.E10 E n = 1 2 n ⁢ ∑ k = 1 n + 1 ( n + 1 k ) ⁢ ∑ j = 0 k - 1 ( - 1 ) j ⁢ ( 2 ⁢ j + 1 ) n Euler-number-E 𝑛 1 superscript 2 𝑛 superscript subscript 𝑘 1 𝑛 1 binomial 𝑛 1 𝑘 superscript subscript 𝑗 0 𝑘 1 superscript 1 𝑗 superscript 2 𝑗 1 𝑛 {\displaystyle{\displaystyle 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}}}
\EulernumberE{n} = \frac{1}{2^{n}}\sum_{k=1}^{n+1}{n+1\choose k}\sum_{j=0}^{k-1}(-1)^{j}(2j+1)^{n}		 

euler(n) = (1)/((2)^(n))*sum(binomial(n + 1,k)*sum((- 1)^(j)*(2*j + 1)^(n), j = 0..k - 1), k = 1..n + 1)

EulerE[n] == Divide[1,(2)^(n)]*Sum[Binomial[n + 1,k]*Sum[(- 1)^(j)*(2*j + 1)^(n), {j, 0, k - 1}, GenerateConditions->None], {k, 1, n + 1}, GenerateConditions->None]
Missing Macro Error Failure - Successful [Tested: 3]
24.6.E11 B n = n 2 n ⁢ ( 2 n - 1 ) ⁢ ∑ k = 1 n ∑ j = 0 k - 1 ( - 1 ) j + 1 ⁢ ( n k ) ⁢ j n - 1 Bernoulli-number-B 𝑛 𝑛 superscript 2 𝑛 superscript 2 𝑛 1 superscript subscript 𝑘 1 𝑛 superscript subscript 𝑗 0 𝑘 1 superscript 1 𝑗 1 binomial 𝑛 𝑘 superscript 𝑗 𝑛 1 {\displaystyle{\displaystyle 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}}}
\BernoullinumberB{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}		 

bernoulli(n) = (n)/((2)^(n)*((2)^(n)- 1))*sum(sum((- 1)^(j + 1)*binomial(n,k)*(j)^(n - 1), j = 0..k - 1), k = 1..n)

BernoulliB[n] == Divide[n,(2)^(n)*((2)^(n)- 1)]*Sum[Sum[(- 1)^(j + 1)*Binomial[n,k]*(j)^(n - 1), {j, 0, k - 1}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]
Aborted Aborted Successful [Tested: 3] Skipped - Because timed out
24.6.E12 E 2 ⁢ n = ∑ k = 0 2 ⁢ n 1 2 k ⁢ ∑ j = 0 k ( - 1 ) j ⁢ ( k j ) ⁢ ( 1 + 2 ⁢ j ) 2 ⁢ n Euler-number-E 2 𝑛 superscript subscript 𝑘 0 2 𝑛 1 superscript 2 𝑘 superscript subscript 𝑗 0 𝑘 superscript 1 𝑗 binomial 𝑘 𝑗 superscript 1 2 𝑗 2 𝑛 {\displaystyle{\displaystyle E_{2n}=\sum_{k=0}^{2n}\frac{1}{2^{k}}\sum_{j=0}^{% k}(-1)^{j}{k\choose j}(1+2j)^{2n}}}
\EulernumberE{2n} = \sum_{k=0}^{2n}\frac{1}{2^{k}}\sum_{j=0}^{k}(-1)^{j}{k\choose j}(1+2j)^{2n}		 

euler(2*n) = sum((1)/((2)^(k))*sum((- 1)^(j)*binomial(k,j)*(1 + 2*j)^(2*n), j = 0..k), k = 0..2*n)

EulerE[2*n] == Sum[Divide[1,(2)^(k)]*Sum[(- 1)^(j)*Binomial[k,j]*(1 + 2*j)^(2*n), {j, 0, k}, GenerateConditions->None], {k, 0, 2*n}, GenerateConditions->None]
Missing Macro Error Failure - Successful [Tested: 3]
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