24.4: Difference between revisions

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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/24.4.E1 24.4.E1] || [[Item:Q7414|<math>\BernoullipolyB{n}@{x+1}-\BernoullipolyB{n}@{x} = nx^{n-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{x+1}-\BernoullipolyB{n}@{x} = nx^{n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, x + 1)- bernoulli(n, x) = n*(x)^(n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, x + 1]- BernoulliB[n, x] == n*(x)^(n - 1)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E1 24.4.E1] || <math qid="Q7414">\BernoullipolyB{n}@{x+1}-\BernoullipolyB{n}@{x} = nx^{n-1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{x+1}-\BernoullipolyB{n}@{x} = nx^{n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, x + 1)- bernoulli(n, x) = n*(x)^(n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, x + 1]- BernoulliB[n, x] == n*(x)^(n - 1)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E2 24.4.E2] || [[Item:Q7415|<math>\EulerpolyE{n}@{x+1}+\EulerpolyE{n}@{x} = 2x^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{x+1}+\EulerpolyE{n}@{x} = 2x^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, x + 1)+ euler(n, x) = 2*(x)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, x + 1]+ EulerE[n, x] == 2*(x)^(n)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E2 24.4.E2] || <math qid="Q7415">\EulerpolyE{n}@{x+1}+\EulerpolyE{n}@{x} = 2x^{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{x+1}+\EulerpolyE{n}@{x} = 2x^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, x + 1)+ euler(n, x) = 2*(x)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, x + 1]+ EulerE[n, x] == 2*(x)^(n)</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E3 24.4.E3] || [[Item:Q7416|<math>\BernoullipolyB{n}@{1-x} = (-1)^{n}\BernoullipolyB{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{1-x} = (-1)^{n}\BernoullipolyB{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, 1 - x) = (- 1)^(n)* bernoulli(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, 1 - x] == (- 1)^(n)* BernoulliB[n, x]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E3 24.4.E3] || <math qid="Q7416">\BernoullipolyB{n}@{1-x} = (-1)^{n}\BernoullipolyB{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{1-x} = (-1)^{n}\BernoullipolyB{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, 1 - x) = (- 1)^(n)* bernoulli(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, 1 - x] == (- 1)^(n)* BernoulliB[n, x]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E4 24.4.E4] || [[Item:Q7417|<math>\EulerpolyE{n}@{1-x} = (-1)^{n}\EulerpolyE{n}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{1-x} = (-1)^{n}\EulerpolyE{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, 1 - x) = (- 1)^(n)* euler(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, 1 - x] == (- 1)^(n)* EulerE[n, x]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E4 24.4.E4] || <math qid="Q7417">\EulerpolyE{n}@{1-x} = (-1)^{n}\EulerpolyE{n}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{1-x} = (-1)^{n}\EulerpolyE{n}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, 1 - x) = (- 1)^(n)* euler(n, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, 1 - x] == (- 1)^(n)* EulerE[n, x]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E5 24.4.E5] || [[Item:Q7418|<math>(-1)^{n}\BernoullipolyB{n}@{-x} = \BernoullipolyB{n}@{x}+nx^{n-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n}\BernoullipolyB{n}@{-x} = \BernoullipolyB{n}@{x}+nx^{n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(n)* bernoulli(n, - x) = bernoulli(n, x)+ n*(x)^(n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n)* BernoulliB[n, - x] == BernoulliB[n, x]+ n*(x)^(n - 1)</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E5 24.4.E5] || <math qid="Q7418">(-1)^{n}\BernoullipolyB{n}@{-x} = \BernoullipolyB{n}@{x}+nx^{n-1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n}\BernoullipolyB{n}@{-x} = \BernoullipolyB{n}@{x}+nx^{n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(n)* bernoulli(n, - x) = bernoulli(n, x)+ n*(x)^(n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n)* BernoulliB[n, - x] == BernoulliB[n, x]+ n*(x)^(n - 1)</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E6 24.4.E6] || [[Item:Q7419|<math>(-1)^{n+1}\EulerpolyE{n}@{-x} = \EulerpolyE{n}@{x}-2x^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n+1}\EulerpolyE{n}@{-x} = \EulerpolyE{n}@{x}-2x^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(n + 1)* euler(n, - x) = euler(n, x)- 2*(x)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n + 1)* EulerE[n, - x] == EulerE[n, x]- 2*(x)^(n)</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E6 24.4.E6] || <math qid="Q7419">(-1)^{n+1}\EulerpolyE{n}@{-x} = \EulerpolyE{n}@{x}-2x^{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n+1}\EulerpolyE{n}@{-x} = \EulerpolyE{n}@{x}-2x^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(n + 1)* euler(n, - x) = euler(n, x)- 2*(x)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n + 1)* EulerE[n, - x] == EulerE[n, x]- 2*(x)^(n)</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E7 24.4.E7] || [[Item:Q7420|<math>\sum_{k=1}^{m}k^{n} = \frac{\BernoullipolyB{n+1}@{m+1}-\BernoullinumberB{n+1}}{n+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=1}^{m}k^{n} = \frac{\BernoullipolyB{n+1}@{m+1}-\BernoullinumberB{n+1}}{n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((k)^(n), k = 1..m) = (bernoulli(n + 1, m + 1)- bernoulli(n + 1))/(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(k)^(n), {k, 1, m}, GenerateConditions->None] == Divide[BernoulliB[n + 1, m + 1]- BernoulliB[n + 1],n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E7 24.4.E7] || <math qid="Q7420">\sum_{k=1}^{m}k^{n} = \frac{\BernoullipolyB{n+1}@{m+1}-\BernoullinumberB{n+1}}{n+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=1}^{m}k^{n} = \frac{\BernoullipolyB{n+1}@{m+1}-\BernoullinumberB{n+1}}{n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((k)^(n), k = 1..m) = (bernoulli(n + 1, m + 1)- bernoulli(n + 1))/(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(k)^(n), {k, 1, m}, GenerateConditions->None] == Divide[BernoulliB[n + 1, m + 1]- BernoulliB[n + 1],n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E8 24.4.E8] || [[Item:Q7421|<math>\sum_{k=1}^{m}(-1)^{m-k}k^{n} = \frac{\EulerpolyE{n}@{m+1}+(-1)^{m}\EulerpolyE{n}@{0}}{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=1}^{m}(-1)^{m-k}k^{n} = \frac{\EulerpolyE{n}@{m+1}+(-1)^{m}\EulerpolyE{n}@{0}}{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((- 1)^(m - k)* (k)^(n), k = 1..m) = (euler(n, m + 1)+(- 1)^(m)* euler(n, 0))/(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(m - k)* (k)^(n), {k, 1, m}, GenerateConditions->None] == Divide[EulerE[n, m + 1]+(- 1)^(m)* EulerE[n, 0],2]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E8 24.4.E8] || <math qid="Q7421">\sum_{k=1}^{m}(-1)^{m-k}k^{n} = \frac{\EulerpolyE{n}@{m+1}+(-1)^{m}\EulerpolyE{n}@{0}}{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=1}^{m}(-1)^{m-k}k^{n} = \frac{\EulerpolyE{n}@{m+1}+(-1)^{m}\EulerpolyE{n}@{0}}{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((- 1)^(m - k)* (k)^(n), k = 1..m) = (euler(n, m + 1)+(- 1)^(m)* euler(n, 0))/(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(m - k)* (k)^(n), {k, 1, m}, GenerateConditions->None] == Divide[EulerE[n, m + 1]+(- 1)^(m)* EulerE[n, 0],2]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E9 24.4.E9] || [[Item:Q7422|<math>\sum_{k=0}^{m-1}(a+dk)^{n} = {\frac{d^{n}}{n+1}\left(\BernoullipolyB{n+1}@{m+\frac{a}{d}}-\BernoullipolyB{n+1}@{\frac{a}{d}}\right)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{m-1}(a+dk)^{n} = {\frac{d^{n}}{n+1}\left(\BernoullipolyB{n+1}@{m+\frac{a}{d}}-\BernoullipolyB{n+1}@{\frac{a}{d}}\right)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((a + d*k)^(n), k = 0..m - 1) = ((d)^(n))/(n + 1)*(bernoulli(n + 1, m +(a)/(d))- bernoulli(n + 1, (a)/(d)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(a + d*k)^(n), {k, 0, m - 1}, GenerateConditions->None] == Divide[(d)^(n),n + 1]*(BernoulliB[n + 1, m +Divide[a,d]]- BernoulliB[n + 1, Divide[a,d]])</syntaxhighlight> || Failure || Failure || Successful [Tested: 300] || Successful [Tested: 300]
| [https://dlmf.nist.gov/24.4.E9 24.4.E9] || <math qid="Q7422">\sum_{k=0}^{m-1}(a+dk)^{n} = {\frac{d^{n}}{n+1}\left(\BernoullipolyB{n+1}@{m+\frac{a}{d}}-\BernoullipolyB{n+1}@{\frac{a}{d}}\right)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{m-1}(a+dk)^{n} = {\frac{d^{n}}{n+1}\left(\BernoullipolyB{n+1}@{m+\frac{a}{d}}-\BernoullipolyB{n+1}@{\frac{a}{d}}\right)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((a + d*k)^(n), k = 0..m - 1) = ((d)^(n))/(n + 1)*(bernoulli(n + 1, m +(a)/(d))- bernoulli(n + 1, (a)/(d)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(a + d*k)^(n), {k, 0, m - 1}, GenerateConditions->None] == Divide[(d)^(n),n + 1]*(BernoulliB[n + 1, m +Divide[a,d]]- BernoulliB[n + 1, Divide[a,d]])</syntaxhighlight> || Failure || Failure || Successful [Tested: 300] || Successful [Tested: 300]
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| [https://dlmf.nist.gov/24.4.E10 24.4.E10] || [[Item:Q7423|<math>\sum_{k=0}^{m-1}(-1)^{k}(a+dk)^{n} = {\frac{d^{n}}{2}\left((-1)^{m-1}\EulerpolyE{n}@{m+\frac{a}{d}}+\EulerpolyE{n}@{\frac{a}{d}}\right)}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{m-1}(-1)^{k}(a+dk)^{n} = {\frac{d^{n}}{2}\left((-1)^{m-1}\EulerpolyE{n}@{m+\frac{a}{d}}+\EulerpolyE{n}@{\frac{a}{d}}\right)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((- 1)^(k)*(a + d*k)^(n), k = 0..m - 1) = ((d)^(n))/(2)*((- 1)^(m - 1)* euler(n, m +(a)/(d))+ euler(n, (a)/(d)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(k)*(a + d*k)^(n), {k, 0, m - 1}, GenerateConditions->None] == Divide[(d)^(n),2]*((- 1)^(m - 1)* EulerE[n, m +Divide[a,d]]+ EulerE[n, Divide[a,d]])</syntaxhighlight> || Failure || Failure || Successful [Tested: 300] || Successful [Tested: 300]
| [https://dlmf.nist.gov/24.4.E10 24.4.E10] || <math qid="Q7423">\sum_{k=0}^{m-1}(-1)^{k}(a+dk)^{n} = {\frac{d^{n}}{2}\left((-1)^{m-1}\EulerpolyE{n}@{m+\frac{a}{d}}+\EulerpolyE{n}@{\frac{a}{d}}\right)}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\sum_{k=0}^{m-1}(-1)^{k}(a+dk)^{n} = {\frac{d^{n}}{2}\left((-1)^{m-1}\EulerpolyE{n}@{m+\frac{a}{d}}+\EulerpolyE{n}@{\frac{a}{d}}\right)}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>sum((- 1)^(k)*(a + d*k)^(n), k = 0..m - 1) = ((d)^(n))/(2)*((- 1)^(m - 1)* euler(n, m +(a)/(d))+ euler(n, (a)/(d)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Sum[(- 1)^(k)*(a + d*k)^(n), {k, 0, m - 1}, GenerateConditions->None] == Divide[(d)^(n),2]*((- 1)^(m - 1)* EulerE[n, m +Divide[a,d]]+ EulerE[n, Divide[a,d]])</syntaxhighlight> || Failure || Failure || Successful [Tested: 300] || Successful [Tested: 300]
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| [https://dlmf.nist.gov/24.4.E12 24.4.E12] || [[Item:Q7425|<math>\BernoullipolyB{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\BernoullipolyB{k}@{x}h^{n-k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\BernoullipolyB{k}@{x}h^{n-k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, x + h) = sum(binomial(n,k)*bernoulli(k, x)*(h)^(n - k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, x + h] == Sum[Binomial[n,k]*BernoulliB[k, x]*(h)^(n - k), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 90] || Successful [Tested: 90]
| [https://dlmf.nist.gov/24.4.E12 24.4.E12] || <math qid="Q7425">\BernoullipolyB{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\BernoullipolyB{k}@{x}h^{n-k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\BernoullipolyB{k}@{x}h^{n-k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, x + h) = sum(binomial(n,k)*bernoulli(k, x)*(h)^(n - k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, x + h] == Sum[Binomial[n,k]*BernoulliB[k, x]*(h)^(n - k), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 90] || Successful [Tested: 90]
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| [https://dlmf.nist.gov/24.4.E13 24.4.E13] || [[Item:Q7426|<math>\EulerpolyE{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\EulerpolyE{k}@{x}h^{n-k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\EulerpolyE{k}@{x}h^{n-k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, x + h) = sum(binomial(n,k)*euler(k, x)*(h)^(n - k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, x + h] == Sum[Binomial[n,k]*EulerE[k, x]*(h)^(n - k), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 90] || Successful [Tested: 90]
| [https://dlmf.nist.gov/24.4.E13 24.4.E13] || <math qid="Q7426">\EulerpolyE{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\EulerpolyE{k}@{x}h^{n-k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\EulerpolyE{k}@{x}h^{n-k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, x + h) = sum(binomial(n,k)*euler(k, x)*(h)^(n - k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, x + h] == Sum[Binomial[n,k]*EulerE[k, x]*(h)^(n - k), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 90] || Successful [Tested: 90]
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| [https://dlmf.nist.gov/24.4.E14 24.4.E14] || [[Item:Q7427|<math>\EulerpolyE{n-1}@{x} = \frac{2}{n}\sum_{k=0}^{n}{n\choose k}(1-2^{k})\BernoullinumberB{k}x^{n-k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n-1}@{x} = \frac{2}{n}\sum_{k=0}^{n}{n\choose k}(1-2^{k})\BernoullinumberB{k}x^{n-k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n - 1, x) = (2)/(n)*sum(binomial(n,k)*(1 - (2)^(k))*bernoulli(k)*(x)^(n - k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n - 1, x] == Divide[2,n]*Sum[Binomial[n,k]*(1 - (2)^(k))*BernoulliB[k]*(x)^(n - k), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E14 24.4.E14] || <math qid="Q7427">\EulerpolyE{n-1}@{x} = \frac{2}{n}\sum_{k=0}^{n}{n\choose k}(1-2^{k})\BernoullinumberB{k}x^{n-k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n-1}@{x} = \frac{2}{n}\sum_{k=0}^{n}{n\choose k}(1-2^{k})\BernoullinumberB{k}x^{n-k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n - 1, x) = (2)/(n)*sum(binomial(n,k)*(1 - (2)^(k))*bernoulli(k)*(x)^(n - k), k = 0..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n - 1, x] == Divide[2,n]*Sum[Binomial[n,k]*(1 - (2)^(k))*BernoulliB[k]*(x)^(n - k), {k, 0, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E15 24.4.E15] || [[Item:Q7428|<math>\BernoullinumberB{2n} = \frac{2n}{2^{2n}(2^{2n}-1)}\sum_{k=0}^{n-1}{2n-1\choose 2k}\EulernumberE{2k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullinumberB{2n} = \frac{2n}{2^{2n}(2^{2n}-1)}\sum_{k=0}^{n-1}{2n-1\choose 2k}\EulernumberE{2k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(2*n) = (2*n)/((2)^(2*n)*((2)^(2*n)- 1))*sum(binomial(2*n - 1,2*k)*euler(2*k), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[2*n] == Divide[2*n,(2)^(2*n)*((2)^(2*n)- 1)]*Sum[Binomial[2*n - 1,2*k]*EulerE[2*k], {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E15 24.4.E15] || <math qid="Q7428">\BernoullinumberB{2n} = \frac{2n}{2^{2n}(2^{2n}-1)}\sum_{k=0}^{n-1}{2n-1\choose 2k}\EulernumberE{2k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullinumberB{2n} = \frac{2n}{2^{2n}(2^{2n}-1)}\sum_{k=0}^{n-1}{2n-1\choose 2k}\EulernumberE{2k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(2*n) = (2*n)/((2)^(2*n)*((2)^(2*n)- 1))*sum(binomial(2*n - 1,2*k)*euler(2*k), k = 0..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[2*n] == Divide[2*n,(2)^(2*n)*((2)^(2*n)- 1)]*Sum[Binomial[2*n - 1,2*k]*EulerE[2*k], {k, 0, n - 1}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E16 24.4.E16] || [[Item:Q7429|<math>\EulernumberE{2n} = \frac{1}{2n+1}-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k-1}-1)\BernoullinumberB{2k}}{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulernumberE{2n} = \frac{1}{2n+1}-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k-1}-1)\BernoullinumberB{2k}}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(2*n) = (1)/(2*n + 1)- sum(binomial(2*n,2*k - 1)*((2)^(2*k)*((2)^(2*k - 1)- 1)*bernoulli(2*k))/(k), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[2*n] == Divide[1,2*n + 1]- Sum[Binomial[2*n,2*k - 1]*Divide[(2)^(2*k)*((2)^(2*k - 1)- 1)*BernoulliB[2*k],k], {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
| [https://dlmf.nist.gov/24.4.E16 24.4.E16] || <math qid="Q7429">\EulernumberE{2n} = \frac{1}{2n+1}-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k-1}-1)\BernoullinumberB{2k}}{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulernumberE{2n} = \frac{1}{2n+1}-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k-1}-1)\BernoullinumberB{2k}}{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(2*n) = (1)/(2*n + 1)- sum(binomial(2*n,2*k - 1)*((2)^(2*k)*((2)^(2*k - 1)- 1)*bernoulli(2*k))/(k), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[2*n] == Divide[1,2*n + 1]- Sum[Binomial[2*n,2*k - 1]*Divide[(2)^(2*k)*((2)^(2*k - 1)- 1)*BernoulliB[2*k],k], {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/24.4.E17 24.4.E17] || [[Item:Q7430|<math>\EulernumberE{2n} = 1-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k}-1)\BernoullinumberB{2k}}{2k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulernumberE{2n} = 1-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k}-1)\BernoullinumberB{2k}}{2k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(2*n) = 1 - sum(binomial(2*n,2*k - 1)*((2)^(2*k)*((2)^(2*k)- 1)*bernoulli(2*k))/(2*k), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[2*n] == 1 - Sum[Binomial[2*n,2*k - 1]*Divide[(2)^(2*k)*((2)^(2*k)- 1)*BernoulliB[2*k],2*k], {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
| [https://dlmf.nist.gov/24.4.E17 24.4.E17] || <math qid="Q7430">\EulernumberE{2n} = 1-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k}-1)\BernoullinumberB{2k}}{2k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulernumberE{2n} = 1-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k}-1)\BernoullinumberB{2k}}{2k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(2*n) = 1 - sum(binomial(2*n,2*k - 1)*((2)^(2*k)*((2)^(2*k)- 1)*bernoulli(2*k))/(2*k), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[2*n] == 1 - Sum[Binomial[2*n,2*k - 1]*Divide[(2)^(2*k)*((2)^(2*k)- 1)*BernoulliB[2*k],2*k], {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Aborted || - || Skipped - Because timed out
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| [https://dlmf.nist.gov/24.4.E18 24.4.E18] || [[Item:Q7431|<math>\BernoullipolyB{n}@{mx} = m^{n-1}\sum_{k=0}^{m-1}\BernoullipolyB{n}@{x+\frac{k}{m}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{mx} = m^{n-1}\sum_{k=0}^{m-1}\BernoullipolyB{n}@{x+\frac{k}{m}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, m*x) = (m)^(n - 1)* sum(bernoulli(n, x +(k)/(m)), k = 0..m - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, m*x] == (m)^(n - 1)* Sum[BernoulliB[n, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 27] || Successful [Tested: 27]
| [https://dlmf.nist.gov/24.4.E18 24.4.E18] || <math qid="Q7431">\BernoullipolyB{n}@{mx} = m^{n-1}\sum_{k=0}^{m-1}\BernoullipolyB{n}@{x+\frac{k}{m}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{mx} = m^{n-1}\sum_{k=0}^{m-1}\BernoullipolyB{n}@{x+\frac{k}{m}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, m*x) = (m)^(n - 1)* sum(bernoulli(n, x +(k)/(m)), k = 0..m - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, m*x] == (m)^(n - 1)* Sum[BernoulliB[n, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 27] || Successful [Tested: 27]
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| [https://dlmf.nist.gov/24.4.E19 24.4.E19] || [[Item:Q7432|<math>\EulerpolyE{n}@{mx} = -\frac{2m^{n}}{n+1}\sum_{k=0}^{m-1}(-1)^{k}\BernoullipolyB{n+1}@{x+\frac{k}{m}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{mx} = -\frac{2m^{n}}{n+1}\sum_{k=0}^{m-1}(-1)^{k}\BernoullipolyB{n+1}@{x+\frac{k}{m}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, m*x) = -(2*(m)^(n))/(n + 1)*sum((- 1)^(k)* bernoulli(n + 1, x +(k)/(m)), k = 0..m - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, m*x] == -Divide[2*(m)^(n),n + 1]*Sum[(- 1)^(k)* BernoulliB[n + 1, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.9166666666666667
| [https://dlmf.nist.gov/24.4.E19 24.4.E19] || <math qid="Q7432">\EulerpolyE{n}@{mx} = -\frac{2m^{n}}{n+1}\sum_{k=0}^{m-1}(-1)^{k}\BernoullipolyB{n+1}@{x+\frac{k}{m}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{mx} = -\frac{2m^{n}}{n+1}\sum_{k=0}^{m-1}(-1)^{k}\BernoullipolyB{n+1}@{x+\frac{k}{m}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, m*x) = -(2*(m)^(n))/(n + 1)*sum((- 1)^(k)* bernoulli(n + 1, x +(k)/(m)), k = 0..m - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, m*x] == -Divide[2*(m)^(n),n + 1]*Sum[(- 1)^(k)* BernoulliB[n + 1, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.9166666666666667
Test Values: {Rule[m, 1], Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.25
Test Values: {Rule[m, 1], Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.25
Test Values: {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/24.4.E20 24.4.E20] || [[Item:Q7433|<math>\EulerpolyE{n}@{mx} = m^{n}\sum_{k=0}^{m-1}(-1)^{k}\EulerpolyE{n}@{x+\frac{k}{m}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{mx} = m^{n}\sum_{k=0}^{m-1}(-1)^{k}\EulerpolyE{n}@{x+\frac{k}{m}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, m*x) = (m)^(n)* sum((- 1)^(k)* euler(n, x +(k)/(m)), k = 0..m - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, m*x] == (m)^(n)* Sum[(- 1)^(k)* EulerE[n, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.5
| [https://dlmf.nist.gov/24.4.E20 24.4.E20] || <math qid="Q7433">\EulerpolyE{n}@{mx} = m^{n}\sum_{k=0}^{m-1}(-1)^{k}\EulerpolyE{n}@{x+\frac{k}{m}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{mx} = m^{n}\sum_{k=0}^{m-1}(-1)^{k}\EulerpolyE{n}@{x+\frac{k}{m}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, m*x) = (m)^(n)* sum((- 1)^(k)* euler(n, x +(k)/(m)), k = 0..m - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, m*x] == (m)^(n)* Sum[(- 1)^(k)* EulerE[n, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || <div class="toccolours mw-collapsible mw-collapsed">Failed [9 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 3.5
Test Values: {Rule[m, 2], Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 11.0
Test Values: {Rule[m, 2], Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 11.0
Test Values: {Rule[m, 2], Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[m, 2], Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/24.4.E21 24.4.E21] || [[Item:Q7434|<math>\BernoullipolyB{n}@{x} = 2^{n-1}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x}+\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{x} = 2^{n-1}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x}+\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, x) = (2)^(n - 1)*(bernoulli(n, (1)/(2)*x)+ bernoulli(n, (1)/(2)*x +(1)/(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, x] == (2)^(n - 1)*(BernoulliB[n, Divide[1,2]*x]+ BernoulliB[n, Divide[1,2]*x +Divide[1,2]])</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E21 24.4.E21] || <math qid="Q7434">\BernoullipolyB{n}@{x} = 2^{n-1}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x}+\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{x} = 2^{n-1}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x}+\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, x) = (2)^(n - 1)*(bernoulli(n, (1)/(2)*x)+ bernoulli(n, (1)/(2)*x +(1)/(2)))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, x] == (2)^(n - 1)*(BernoulliB[n, Divide[1,2]*x]+ BernoulliB[n, Divide[1,2]*x +Divide[1,2]])</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E22 24.4.E22] || [[Item:Q7435|<math>\EulerpolyE{n-1}@{x} = \frac{2}{n}\left(\BernoullipolyB{n}@{x}-2^{n}\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n-1}@{x} = \frac{2}{n}\left(\BernoullipolyB{n}@{x}-2^{n}\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n - 1, x) = (2)/(n)*(bernoulli(n, x)- (2)^(n)* bernoulli(n, (1)/(2)*x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n - 1, x] == Divide[2,n]*(BernoulliB[n, x]- (2)^(n)* BernoulliB[n, Divide[1,2]*x])</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E22 24.4.E22] || <math qid="Q7435">\EulerpolyE{n-1}@{x} = \frac{2}{n}\left(\BernoullipolyB{n}@{x}-2^{n}\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n-1}@{x} = \frac{2}{n}\left(\BernoullipolyB{n}@{x}-2^{n}\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n - 1, x) = (2)/(n)*(bernoulli(n, x)- (2)^(n)* bernoulli(n, (1)/(2)*x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n - 1, x] == Divide[2,n]*(BernoulliB[n, x]- (2)^(n)* BernoulliB[n, Divide[1,2]*x])</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E23 24.4.E23] || [[Item:Q7436|<math>\EulerpolyE{n-1}@{x} = \frac{2^{n}}{n}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}-\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n-1}@{x} = \frac{2^{n}}{n}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}-\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n - 1, x) = ((2)^(n))/(n)*(bernoulli(n, (1)/(2)*x +(1)/(2))- bernoulli(n, (1)/(2)*x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n - 1, x] == Divide[(2)^(n),n]*(BernoulliB[n, Divide[1,2]*x +Divide[1,2]]- BernoulliB[n, Divide[1,2]*x])</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
| [https://dlmf.nist.gov/24.4.E23 24.4.E23] || <math qid="Q7436">\EulerpolyE{n-1}@{x} = \frac{2^{n}}{n}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}-\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n-1}@{x} = \frac{2^{n}}{n}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}-\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n - 1, x) = ((2)^(n))/(n)*(bernoulli(n, (1)/(2)*x +(1)/(2))- bernoulli(n, (1)/(2)*x))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n - 1, x] == Divide[(2)^(n),n]*(BernoulliB[n, Divide[1,2]*x +Divide[1,2]]- BernoulliB[n, Divide[1,2]*x])</syntaxhighlight> || Failure || Failure || Successful [Tested: 9] || Successful [Tested: 9]
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| [https://dlmf.nist.gov/24.4.E24 24.4.E24] || [[Item:Q7437|<math>\BernoullipolyB{n}@{mx} = m^{n}\BernoullipolyB{n}@{x}+n\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}\frac{e^{2\pi i(k-j)r/m}}{(1-e^{2\pi ir/m})^{n}}\right)(j+mx)^{n-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{mx} = m^{n}\BernoullipolyB{n}@{x}+n\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}\frac{e^{2\pi i(k-j)r/m}}{(1-e^{2\pi ir/m})^{n}}\right)(j+mx)^{n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, m*x) = (m)^(n)* bernoulli(n, x)+ n*sum(sum((- 1)^(j)*binomial(n,k)*(sum((exp(2*Pi*I*(k - j)*r/m))/((1 - exp(2*Pi*I*r/m))^(n)), r = 1..m - 1))*(j + m*x)^(n - 1), j = 0..k - 1), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, m*x] == (m)^(n)* BernoulliB[n, x]+ n*Sum[Sum[(- 1)^(j)*Binomial[n,k]*(Sum[Divide[Exp[2*Pi*I*(k - j)*r/m],(1 - Exp[2*Pi*I*r/m])^(n)], {r, 1, m - 1}, GenerateConditions->None])*(j + m*x)^(n - 1), {j, 0, k - 1}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [17 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.0
| [https://dlmf.nist.gov/24.4.E24 24.4.E24] || <math qid="Q7437">\BernoullipolyB{n}@{mx} = m^{n}\BernoullipolyB{n}@{x}+n\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}\frac{e^{2\pi i(k-j)r/m}}{(1-e^{2\pi ir/m})^{n}}\right)(j+mx)^{n-1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{mx} = m^{n}\BernoullipolyB{n}@{x}+n\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}\frac{e^{2\pi i(k-j)r/m}}{(1-e^{2\pi ir/m})^{n}}\right)(j+mx)^{n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, m*x) = (m)^(n)* bernoulli(n, x)+ n*sum(sum((- 1)^(j)*binomial(n,k)*(sum((exp(2*Pi*I*(k - j)*r/m))/((1 - exp(2*Pi*I*r/m))^(n)), r = 1..m - 1))*(j + m*x)^(n - 1), j = 0..k - 1), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, m*x] == (m)^(n)* BernoulliB[n, x]+ n*Sum[Sum[(- 1)^(j)*Binomial[n,k]*(Sum[Divide[Exp[2*Pi*I*(k - j)*r/m],(1 - Exp[2*Pi*I*r/m])^(n)], {r, 1, m - 1}, GenerateConditions->None])*(j + m*x)^(n - 1), {j, 0, k - 1}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [17 / 27]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 1.0
Test Values: {Rule[m, 2], Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.9999999999999991
Test Values: {Rule[m, 2], Rule[n, 1], Rule[x, 1.5]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: 1.9999999999999991
Test Values: {Rule[m, 2], Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[m, 2], Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/24.4.E25 24.4.E25] || [[Item:Q7438|<math>\BernoullipolyB{n}@{0} = (-1)^{n}\BernoullipolyB{n}@{1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{0} = (-1)^{n}\BernoullipolyB{n}@{1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, 0) = (- 1)^(n)* bernoulli(n, 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, 0] == (- 1)^(n)* BernoulliB[n, 1]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E25 24.4.E25] || <math qid="Q7438">\BernoullipolyB{n}@{0} = (-1)^{n}\BernoullipolyB{n}@{1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{0} = (-1)^{n}\BernoullipolyB{n}@{1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, 0) = (- 1)^(n)* bernoulli(n, 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, 0] == (- 1)^(n)* BernoulliB[n, 1]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E25 24.4.E25] || [[Item:Q7438|<math>(-1)^{n}\BernoullipolyB{n}@{1} = \BernoullinumberB{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n}\BernoullipolyB{n}@{1} = \BernoullinumberB{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(n)* bernoulli(n, 1) = bernoulli(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n)* BernoulliB[n, 1] == BernoulliB[n]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E25 24.4.E25] || <math qid="Q7438">(-1)^{n}\BernoullipolyB{n}@{1} = \BernoullinumberB{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n}\BernoullipolyB{n}@{1} = \BernoullinumberB{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(n)* bernoulli(n, 1) = bernoulli(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n)* BernoulliB[n, 1] == BernoulliB[n]</syntaxhighlight> || Successful || Failure || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E26 24.4.E26] || [[Item:Q7439|<math>\EulerpolyE{n}@{0} = -\EulerpolyE{n}@{1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{0} = -\EulerpolyE{n}@{1}</syntaxhighlight> || <math>n > 0</math> || <syntaxhighlight lang=mathematica>euler(n, 0) = - euler(n, 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, 0] == - EulerE[n, 1]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E26 24.4.E26] || <math qid="Q7439">\EulerpolyE{n}@{0} = -\EulerpolyE{n}@{1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{0} = -\EulerpolyE{n}@{1}</syntaxhighlight> || <math>n > 0</math> || <syntaxhighlight lang=mathematica>euler(n, 0) = - euler(n, 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, 0] == - EulerE[n, 1]</syntaxhighlight> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E26 24.4.E26] || [[Item:Q7439|<math>-\EulerpolyE{n}@{1} = -\frac{2}{n+1}(2^{n+1}-1)\BernoullinumberB{n+1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\EulerpolyE{n}@{1} = -\frac{2}{n+1}(2^{n+1}-1)\BernoullinumberB{n+1}</syntaxhighlight> || <math>n > 0</math> || <syntaxhighlight lang=mathematica>- euler(n, 1) = -(2)/(n + 1)*((2)^(n + 1)- 1)*bernoulli(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- EulerE[n, 1] == -Divide[2,n + 1]*((2)^(n + 1)- 1)*BernoulliB[n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E26 24.4.E26] || <math qid="Q7439">-\EulerpolyE{n}@{1} = -\frac{2}{n+1}(2^{n+1}-1)\BernoullinumberB{n+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\EulerpolyE{n}@{1} = -\frac{2}{n+1}(2^{n+1}-1)\BernoullinumberB{n+1}</syntaxhighlight> || <math>n > 0</math> || <syntaxhighlight lang=mathematica>- euler(n, 1) = -(2)/(n + 1)*((2)^(n + 1)- 1)*bernoulli(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- EulerE[n, 1] == -Divide[2,n + 1]*((2)^(n + 1)- 1)*BernoulliB[n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E27 24.4.E27] || [[Item:Q7440|<math>\BernoullipolyB{n}@{\tfrac{1}{2}} = -(1-2^{1-n})\BernoullinumberB{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{\tfrac{1}{2}} = -(1-2^{1-n})\BernoullinumberB{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, (1)/(2)) = -(1 - (2)^(1 - n))*bernoulli(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, Divide[1,2]] == -(1 - (2)^(1 - n))*BernoulliB[n]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E27 24.4.E27] || <math qid="Q7440">\BernoullipolyB{n}@{\tfrac{1}{2}} = -(1-2^{1-n})\BernoullinumberB{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{\tfrac{1}{2}} = -(1-2^{1-n})\BernoullinumberB{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, (1)/(2)) = -(1 - (2)^(1 - n))*bernoulli(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, Divide[1,2]] == -(1 - (2)^(1 - n))*BernoulliB[n]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E28 24.4.E28] || [[Item:Q7441|<math>\EulerpolyE{n}@{\tfrac{1}{2}} = 2^{-n}\EulernumberE{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{\tfrac{1}{2}} = 2^{-n}\EulernumberE{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, (1)/(2)) = (2)^(- n)* euler(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, Divide[1,2]] == (2)^(- n)* EulerE[n]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E28 24.4.E28] || <math qid="Q7441">\EulerpolyE{n}@{\tfrac{1}{2}} = 2^{-n}\EulernumberE{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{\tfrac{1}{2}} = 2^{-n}\EulernumberE{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, (1)/(2)) = (2)^(- n)* euler(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, Divide[1,2]] == (2)^(- n)* EulerE[n]</syntaxhighlight> || Missing Macro Error || Successful || - || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E29 24.4.E29] || [[Item:Q7442|<math>\BernoullipolyB{2n}@{\tfrac{1}{3}} = \BernoullipolyB{2n}@{\tfrac{2}{3}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{2n}@{\tfrac{1}{3}} = \BernoullipolyB{2n}@{\tfrac{2}{3}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(2*n, (1)/(3)) = bernoulli(2*n, (2)/(3))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[2*n, Divide[1,3]] == BernoulliB[2*n, Divide[2,3]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E29 24.4.E29] || <math qid="Q7442">\BernoullipolyB{2n}@{\tfrac{1}{3}} = \BernoullipolyB{2n}@{\tfrac{2}{3}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{2n}@{\tfrac{1}{3}} = \BernoullipolyB{2n}@{\tfrac{2}{3}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(2*n, (1)/(3)) = bernoulli(2*n, (2)/(3))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[2*n, Divide[1,3]] == BernoulliB[2*n, Divide[2,3]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E29 24.4.E29] || [[Item:Q7442|<math>\BernoullipolyB{2n}@{\tfrac{2}{3}} = -\tfrac{1}{2}(1-3^{1-2n})\BernoullinumberB{2n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{2n}@{\tfrac{2}{3}} = -\tfrac{1}{2}(1-3^{1-2n})\BernoullinumberB{2n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(2*n, (2)/(3)) = -(1)/(2)*(1 - (3)^(1 - 2*n))*bernoulli(2*n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[2*n, Divide[2,3]] == -Divide[1,2]*(1 - (3)^(1 - 2*n))*BernoulliB[2*n]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E29 24.4.E29] || <math qid="Q7442">\BernoullipolyB{2n}@{\tfrac{2}{3}} = -\tfrac{1}{2}(1-3^{1-2n})\BernoullinumberB{2n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{2n}@{\tfrac{2}{3}} = -\tfrac{1}{2}(1-3^{1-2n})\BernoullinumberB{2n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(2*n, (2)/(3)) = -(1)/(2)*(1 - (3)^(1 - 2*n))*bernoulli(2*n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[2*n, Divide[2,3]] == -Divide[1,2]*(1 - (3)^(1 - 2*n))*BernoulliB[2*n]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E30 24.4.E30] || [[Item:Q7443|<math>\EulerpolyE{2n-1}@{\tfrac{1}{3}} = -\EulerpolyE{2n-1}@{\tfrac{2}{3}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{2n-1}@{\tfrac{1}{3}} = -\EulerpolyE{2n-1}@{\tfrac{2}{3}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(2*n - 1, (1)/(3)) = - euler(2*n - 1, (2)/(3))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[2*n - 1, Divide[1,3]] == - EulerE[2*n - 1, Divide[2,3]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E30 24.4.E30] || <math qid="Q7443">\EulerpolyE{2n-1}@{\tfrac{1}{3}} = -\EulerpolyE{2n-1}@{\tfrac{2}{3}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{2n-1}@{\tfrac{1}{3}} = -\EulerpolyE{2n-1}@{\tfrac{2}{3}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(2*n - 1, (1)/(3)) = - euler(2*n - 1, (2)/(3))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[2*n - 1, Divide[1,3]] == - EulerE[2*n - 1, Divide[2,3]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E30 24.4.E30] || [[Item:Q7443|<math>-\EulerpolyE{2n-1}@{\tfrac{2}{3}} = -\frac{(1-3^{1-2n})(2^{2n}-1)}{2n}\BernoullinumberB{2n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\EulerpolyE{2n-1}@{\tfrac{2}{3}} = -\frac{(1-3^{1-2n})(2^{2n}-1)}{2n}\BernoullinumberB{2n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>- euler(2*n - 1, (2)/(3)) = -((1 - (3)^(1 - 2*n))*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- EulerE[2*n - 1, Divide[2,3]] == -Divide[(1 - (3)^(1 - 2*n))*((2)^(2*n)- 1),2*n]*BernoulliB[2*n]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E30 24.4.E30] || <math qid="Q7443">-\EulerpolyE{2n-1}@{\tfrac{2}{3}} = -\frac{(1-3^{1-2n})(2^{2n}-1)}{2n}\BernoullinumberB{2n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>-\EulerpolyE{2n-1}@{\tfrac{2}{3}} = -\frac{(1-3^{1-2n})(2^{2n}-1)}{2n}\BernoullinumberB{2n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>- euler(2*n - 1, (2)/(3)) = -((1 - (3)^(1 - 2*n))*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>- EulerE[2*n - 1, Divide[2,3]] == -Divide[(1 - (3)^(1 - 2*n))*((2)^(2*n)- 1),2*n]*BernoulliB[2*n]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E31 24.4.E31] || [[Item:Q7444|<math>\BernoullipolyB{n}@{\tfrac{1}{4}} = (-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{\tfrac{1}{4}} = (-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, (1)/(4)) = (- 1)^(n)* bernoulli(n, (3)/(4))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, Divide[1,4]] == (- 1)^(n)* BernoulliB[n, Divide[3,4]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 1] || Successful [Tested: 1]
| [https://dlmf.nist.gov/24.4.E31 24.4.E31] || <math qid="Q7444">\BernoullipolyB{n}@{\tfrac{1}{4}} = (-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{\tfrac{1}{4}} = (-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, (1)/(4)) = (- 1)^(n)* bernoulli(n, (3)/(4))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, Divide[1,4]] == (- 1)^(n)* BernoulliB[n, Divide[3,4]]</syntaxhighlight> || Failure || Successful || Successful [Tested: 1] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/24.4.E31 24.4.E31] || [[Item:Q7444|<math>(-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}} = -\frac{1-2^{1-n}}{2^{n}}\BernoullinumberB{n}-\frac{n}{4^{n}}\EulernumberE{n-1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}} = -\frac{1-2^{1-n}}{2^{n}}\BernoullinumberB{n}-\frac{n}{4^{n}}\EulernumberE{n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(n)* bernoulli(n, (3)/(4)) = -(1 - (2)^(1 - n))/((2)^(n))*bernoulli(n)-(n)/((4)^(n))*euler(n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n)* BernoulliB[n, Divide[3,4]] == -Divide[1 - (2)^(1 - n),(2)^(n)]*BernoulliB[n]-Divide[n,(4)^(n)]*EulerE[n - 1]</syntaxhighlight> || Missing Macro Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E31 24.4.E31] || <math qid="Q7444">(-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}} = -\frac{1-2^{1-n}}{2^{n}}\BernoullinumberB{n}-\frac{n}{4^{n}}\EulernumberE{n-1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}} = -\frac{1-2^{1-n}}{2^{n}}\BernoullinumberB{n}-\frac{n}{4^{n}}\EulernumberE{n-1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(- 1)^(n)* bernoulli(n, (3)/(4)) = -(1 - (2)^(1 - n))/((2)^(n))*bernoulli(n)-(n)/((4)^(n))*euler(n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(n)* BernoulliB[n, Divide[3,4]] == -Divide[1 - (2)^(1 - n),(2)^(n)]*BernoulliB[n]-Divide[n,(4)^(n)]*EulerE[n - 1]</syntaxhighlight> || Missing Macro Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 3]
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| [https://dlmf.nist.gov/24.4.E32 24.4.E32] || [[Item:Q7445|<math>\BernoullipolyB{2n}@{\tfrac{1}{6}} = \BernoullipolyB{2n}@{\tfrac{5}{6}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{2n}@{\tfrac{1}{6}} = \BernoullipolyB{2n}@{\tfrac{5}{6}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(2*n, (1)/(6)) = bernoulli(2*n, (5)/(6))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[2*n, Divide[1,6]] == BernoulliB[2*n, Divide[5,6]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E32 24.4.E32] || <math qid="Q7445">\BernoullipolyB{2n}@{\tfrac{1}{6}} = \BernoullipolyB{2n}@{\tfrac{5}{6}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{2n}@{\tfrac{1}{6}} = \BernoullipolyB{2n}@{\tfrac{5}{6}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(2*n, (1)/(6)) = bernoulli(2*n, (5)/(6))</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[2*n, Divide[1,6]] == BernoulliB[2*n, Divide[5,6]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-  
|-  
| [https://dlmf.nist.gov/24.4.E32 24.4.E32] || [[Item:Q7445|<math>\BernoullipolyB{2n}@{\tfrac{5}{6}} = \tfrac{1}{2}(1-2^{1-2n})(1-3^{1-2n})\BernoullinumberB{2n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{2n}@{\tfrac{5}{6}} = \tfrac{1}{2}(1-2^{1-2n})(1-3^{1-2n})\BernoullinumberB{2n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(2*n, (5)/(6)) = (1)/(2)*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))*bernoulli(2*n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[2*n, Divide[5,6]] == Divide[1,2]*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))*BernoulliB[2*n]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E32 24.4.E32] || <math qid="Q7445">\BernoullipolyB{2n}@{\tfrac{5}{6}} = \tfrac{1}{2}(1-2^{1-2n})(1-3^{1-2n})\BernoullinumberB{2n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{2n}@{\tfrac{5}{6}} = \tfrac{1}{2}(1-2^{1-2n})(1-3^{1-2n})\BernoullinumberB{2n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(2*n, (5)/(6)) = (1)/(2)*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))*bernoulli(2*n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[2*n, Divide[5,6]] == Divide[1,2]*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))*BernoulliB[2*n]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-  
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| [https://dlmf.nist.gov/24.4.E33 24.4.E33] || [[Item:Q7446|<math>\EulerpolyE{2n}@{\tfrac{1}{6}} = \EulerpolyE{2n}@{\tfrac{5}{6}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{2n}@{\tfrac{1}{6}} = \EulerpolyE{2n}@{\tfrac{5}{6}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(2*n, (1)/(6)) = euler(2*n, (5)/(6))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[2*n, Divide[1,6]] == EulerE[2*n, Divide[5,6]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E33 24.4.E33] || <math qid="Q7446">\EulerpolyE{2n}@{\tfrac{1}{6}} = \EulerpolyE{2n}@{\tfrac{5}{6}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{2n}@{\tfrac{1}{6}} = \EulerpolyE{2n}@{\tfrac{5}{6}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(2*n, (1)/(6)) = euler(2*n, (5)/(6))</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[2*n, Divide[1,6]] == EulerE[2*n, Divide[5,6]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
|-  
|-  
| [https://dlmf.nist.gov/24.4.E33 24.4.E33] || [[Item:Q7446|<math>\EulerpolyE{2n}@{\tfrac{5}{6}} = \frac{1+3^{-2n}}{2^{2n+1}}\EulernumberE{2n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{2n}@{\tfrac{5}{6}} = \frac{1+3^{-2n}}{2^{2n+1}}\EulernumberE{2n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(2*n, (5)/(6)) = (1 + (3)^(- 2*n))/((2)^(2*n + 1))*euler(2*n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[2*n, Divide[5,6]] == Divide[1 + (3)^(- 2*n),(2)^(2*n + 1)]*EulerE[2*n]</syntaxhighlight> || Missing Macro Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E33 24.4.E33] || <math qid="Q7446">\EulerpolyE{2n}@{\tfrac{5}{6}} = \frac{1+3^{-2n}}{2^{2n+1}}\EulernumberE{2n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{2n}@{\tfrac{5}{6}} = \frac{1+3^{-2n}}{2^{2n+1}}\EulernumberE{2n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(2*n, (5)/(6)) = (1 + (3)^(- 2*n))/((2)^(2*n + 1))*euler(2*n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[2*n, Divide[5,6]] == Divide[1 + (3)^(- 2*n),(2)^(2*n + 1)]*EulerE[2*n]</syntaxhighlight> || Missing Macro Error || Failure || Skip - symbolical successful subtest || Successful [Tested: 3]
|-  
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| [https://dlmf.nist.gov/24.4.E34 24.4.E34] || [[Item:Q7447|<math>\deriv{}{x}\BernoullipolyB{n}@{x} = n\BernoullipolyB{n-1}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{x}\BernoullipolyB{n}@{x} = n\BernoullipolyB{n-1}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(bernoulli(n, x), x) = n*bernoulli(n - 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[BernoulliB[n, x], x] == n*BernoulliB[n - 1, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E34 24.4.E34] || <math qid="Q7447">\deriv{}{x}\BernoullipolyB{n}@{x} = n\BernoullipolyB{n-1}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{x}\BernoullipolyB{n}@{x} = n\BernoullipolyB{n-1}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(bernoulli(n, x), x) = n*bernoulli(n - 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[BernoulliB[n, x], x] == n*BernoulliB[n - 1, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
|-  
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| [https://dlmf.nist.gov/24.4.E35 24.4.E35] || [[Item:Q7448|<math>\deriv{}{x}\EulerpolyE{n}@{x} = n\EulerpolyE{n-1}@{x}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{x}\EulerpolyE{n}@{x} = n\EulerpolyE{n-1}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(euler(n, x), x) = n*euler(n - 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[EulerE[n, x], x] == n*EulerE[n - 1, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
| [https://dlmf.nist.gov/24.4.E35 24.4.E35] || <math qid="Q7448">\deriv{}{x}\EulerpolyE{n}@{x} = n\EulerpolyE{n-1}@{x}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\deriv{}{x}\EulerpolyE{n}@{x} = n\EulerpolyE{n-1}@{x}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>diff(euler(n, x), x) = n*euler(n - 1, x)</syntaxhighlight> || <syntaxhighlight lang=mathematica>D[EulerE[n, x], x] == n*EulerE[n - 1, x]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 3]
|-  
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| [https://dlmf.nist.gov/24.4.E37 24.4.E37] || [[Item:Q7450|<math>\BernoullipolyB{n}@{x+h} = (B(x)+h)^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{x+h} = (B(x)+h)^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, x + h) = (B(x)+ h)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, x + h] == (B[x]+ h)^(n)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.299038106-.7500000000*I
| [https://dlmf.nist.gov/24.4.E37 24.4.E37] || <math qid="Q7450">\BernoullipolyB{n}@{x+h} = (B(x)+h)^{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\BernoullipolyB{n}@{x+h} = (B(x)+h)^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>bernoulli(n, x + h) = (B(x)+ h)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>BernoulliB[n, x + h] == (B[x]+ h)^(n)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.299038106-.7500000000*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .23717473e-1-3.546633371*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .23717473e-1-3.546633371*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.299038105676658, -0.7499999999999998]
Test Values: {B = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.299038105676658, -0.7499999999999998]
Line 108: Line 108:
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2], Rule[x, 1.5]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/24.4.E38 24.4.E38] || [[Item:Q7451|<math>P(E(x)+1)+P(E(x)) = 2P(x)</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>P(E(x)+1)+P(E(x)) = 2P(x)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">P(E(x)+ 1)+ P(E(x)) = 2*P(x)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">P[E[x]+ 1]+ P[E[x]] == 2*P[x]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/24.4.E38 24.4.E38] || <math qid="Q7451">P(E(x)+1)+P(E(x)) = 2P(x)</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>P(E(x)+1)+P(E(x)) = 2P(x)</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">P(E(x)+ 1)+ P(E(x)) = 2*P(x)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">P[E[x]+ 1]+ P[E[x]] == 2*P[x]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|-  
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| [https://dlmf.nist.gov/24.4.E39 24.4.E39] || [[Item:Q7452|<math>\EulerpolyE{n}@{x+h} = (E(x)+h)^{n}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{x+h} = (E(x)+h)^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, x + h) = (E(x)+ h)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, x + h] == (E[x]+ h)^(n)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.299038106-.7500000000*I
| [https://dlmf.nist.gov/24.4.E39 24.4.E39] || <math qid="Q7452">\EulerpolyE{n}@{x+h} = (E(x)+h)^{n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\EulerpolyE{n}@{x+h} = (E(x)+h)^{n}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>euler(n, x + h) = (E(x)+ h)^(n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>EulerE[n, x + h] == (E[x]+ h)^(n)</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.299038106-.7500000000*I
Test Values: {E = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.142949194-3.546633371*I
Test Values: {E = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -.142949194-3.546633371*I
Test Values: {E = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.299038105676658, -0.7499999999999998]
Test Values: {E = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2}</syntaxhighlight><br>... skip entries to safe data</div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.299038105676658, -0.7499999999999998]

Latest revision as of 12:01, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
24.4.E1 B n ⁡ ( x + 1 ) - B n ⁡ ( x ) = n ⁢ x n - 1 Bernoulli-polynomial-B 𝑛 𝑥 1 Bernoulli-polynomial-B 𝑛 𝑥 𝑛 superscript 𝑥 𝑛 1 {\displaystyle{\displaystyle B_{n}\left(x+1\right)-B_{n}\left(x\right)=nx^{n-1% }}}
\BernoullipolyB{n}@{x+1}-\BernoullipolyB{n}@{x} = nx^{n-1}		 

bernoulli(n, x + 1)- bernoulli(n, x) = n*(x)^(n - 1)

BernoulliB[n, x + 1]- BernoulliB[n, x] == n*(x)^(n - 1)
Successful Successful - Successful [Tested: 9]
24.4.E2 E n ⁡ ( x + 1 ) + E n ⁡ ( x ) = 2 ⁢ x n Euler-polynomial-E 𝑛 𝑥 1 Euler-polynomial-E 𝑛 𝑥 2 superscript 𝑥 𝑛 {\displaystyle{\displaystyle E_{n}\left(x+1\right)+E_{n}\left(x\right)=2x^{n}}}
\EulerpolyE{n}@{x+1}+\EulerpolyE{n}@{x} = 2x^{n}		 

euler(n, x + 1)+ euler(n, x) = 2*(x)^(n)

EulerE[n, x + 1]+ EulerE[n, x] == 2*(x)^(n)
Successful Successful - Successful [Tested: 9]
24.4.E3 B n ⁡ ( 1 - x ) = ( - 1 ) n ⁢ B n ⁡ ( x ) Bernoulli-polynomial-B 𝑛 1 𝑥 superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 𝑥 {\displaystyle{\displaystyle B_{n}\left(1-x\right)=(-1)^{n}B_{n}\left(x\right)}}
\BernoullipolyB{n}@{1-x} = (-1)^{n}\BernoullipolyB{n}@{x}		 

bernoulli(n, 1 - x) = (- 1)^(n)* bernoulli(n, x)

BernoulliB[n, 1 - x] == (- 1)^(n)* BernoulliB[n, x]
Successful Failure - Successful [Tested: 9]
24.4.E4 E n ⁡ ( 1 - x ) = ( - 1 ) n ⁢ E n ⁡ ( x ) Euler-polynomial-E 𝑛 1 𝑥 superscript 1 𝑛 Euler-polynomial-E 𝑛 𝑥 {\displaystyle{\displaystyle E_{n}\left(1-x\right)=(-1)^{n}E_{n}\left(x\right)}}
\EulerpolyE{n}@{1-x} = (-1)^{n}\EulerpolyE{n}@{x}		 

euler(n, 1 - x) = (- 1)^(n)* euler(n, x)

EulerE[n, 1 - x] == (- 1)^(n)* EulerE[n, x]
Successful Failure - Successful [Tested: 9]
24.4.E5 ( - 1 ) n ⁢ B n ⁡ ( - x ) = B n ⁡ ( x ) + n ⁢ x n - 1 superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 𝑥 Bernoulli-polynomial-B 𝑛 𝑥 𝑛 superscript 𝑥 𝑛 1 {\displaystyle{\displaystyle(-1)^{n}B_{n}\left(-x\right)=B_{n}\left(x\right)+% nx^{n-1}}}
(-1)^{n}\BernoullipolyB{n}@{-x} = \BernoullipolyB{n}@{x}+nx^{n-1}		 

(- 1)^(n)* bernoulli(n, - x) = bernoulli(n, x)+ n*(x)^(n - 1)

(- 1)^(n)* BernoulliB[n, - x] == BernoulliB[n, x]+ n*(x)^(n - 1)
Successful Failure - Successful [Tested: 9]
24.4.E6 ( - 1 ) n + 1 ⁢ E n ⁡ ( - x ) = E n ⁡ ( x ) - 2 ⁢ x n superscript 1 𝑛 1 Euler-polynomial-E 𝑛 𝑥 Euler-polynomial-E 𝑛 𝑥 2 superscript 𝑥 𝑛 {\displaystyle{\displaystyle(-1)^{n+1}E_{n}\left(-x\right)=E_{n}\left(x\right)% -2x^{n}}}
(-1)^{n+1}\EulerpolyE{n}@{-x} = \EulerpolyE{n}@{x}-2x^{n}		 

(- 1)^(n + 1)* euler(n, - x) = euler(n, x)- 2*(x)^(n)

(- 1)^(n + 1)* EulerE[n, - x] == EulerE[n, x]- 2*(x)^(n)
Successful Failure - Successful [Tested: 9]
24.4.E7 ∑ k = 1 m k n = B n + 1 ⁡ ( m + 1 ) - B n + 1 n + 1 superscript subscript 𝑘 1 𝑚 superscript 𝑘 𝑛 Bernoulli-polynomial-B 𝑛 1 𝑚 1 Bernoulli-number-B 𝑛 1 𝑛 1 {\displaystyle{\displaystyle\sum_{k=1}^{m}k^{n}=\frac{B_{n+1}\left(m+1\right)-% B_{n+1}}{n+1}}}
\sum_{k=1}^{m}k^{n} = \frac{\BernoullipolyB{n+1}@{m+1}-\BernoullinumberB{n+1}}{n+1}		 

sum((k)^(n), k = 1..m) = (bernoulli(n + 1, m + 1)- bernoulli(n + 1))/(n + 1)

Sum[(k)^(n), {k, 1, m}, GenerateConditions->None] == Divide[BernoulliB[n + 1, m + 1]- BernoulliB[n + 1],n + 1]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
24.4.E8 ∑ k = 1 m ( - 1 ) m - k ⁢ k n = E n ⁡ ( m + 1 ) + ( - 1 ) m ⁢ E n ⁡ ( 0 ) 2 superscript subscript 𝑘 1 𝑚 superscript 1 𝑚 𝑘 superscript 𝑘 𝑛 Euler-polynomial-E 𝑛 𝑚 1 superscript 1 𝑚 Euler-polynomial-E 𝑛 0 2 {\displaystyle{\displaystyle\sum_{k=1}^{m}(-1)^{m-k}k^{n}=\frac{E_{n}\left(m+1% \right)+(-1)^{m}E_{n}\left(0\right)}{2}}}
\sum_{k=1}^{m}(-1)^{m-k}k^{n} = \frac{\EulerpolyE{n}@{m+1}+(-1)^{m}\EulerpolyE{n}@{0}}{2}		 

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

Sum[(- 1)^(m - k)* (k)^(n), {k, 1, m}, GenerateConditions->None] == Divide[EulerE[n, m + 1]+(- 1)^(m)* EulerE[n, 0],2]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
24.4.E9 ∑ k = 0 m - 1 ( a + d ⁢ k ) n = d n n + 1 ⁢ ( B n + 1 ⁡ ( m + a d ) - B n + 1 ⁡ ( a d ) ) superscript subscript 𝑘 0 𝑚 1 superscript 𝑎 𝑑 𝑘 𝑛 superscript 𝑑 𝑛 𝑛 1 Bernoulli-polynomial-B 𝑛 1 𝑚 𝑎 𝑑 Bernoulli-polynomial-B 𝑛 1 𝑎 𝑑 {\displaystyle{\displaystyle\sum_{k=0}^{m-1}(a+dk)^{n}={\frac{d^{n}}{n+1}\left% (B_{n+1}\left(m+\frac{a}{d}\right)-B_{n+1}\left(\frac{a}{d}\right)\right)}}}
\sum_{k=0}^{m-1}(a+dk)^{n} = {\frac{d^{n}}{n+1}\left(\BernoullipolyB{n+1}@{m+\frac{a}{d}}-\BernoullipolyB{n+1}@{\frac{a}{d}}\right)}		 

sum((a + d*k)^(n), k = 0..m - 1) = ((d)^(n))/(n + 1)*(bernoulli(n + 1, m +(a)/(d))- bernoulli(n + 1, (a)/(d)))

Sum[(a + d*k)^(n), {k, 0, m - 1}, GenerateConditions->None] == Divide[(d)^(n),n + 1]*(BernoulliB[n + 1, m +Divide[a,d]]- BernoulliB[n + 1, Divide[a,d]])
Failure Failure Successful [Tested: 300] Successful [Tested: 300]
24.4.E10 ∑ k = 0 m - 1 ( - 1 ) k ⁢ ( a + d ⁢ k ) n = d n 2 ⁢ ( ( - 1 ) m - 1 ⁢ E n ⁡ ( m + a d ) + E n ⁡ ( a d ) ) superscript subscript 𝑘 0 𝑚 1 superscript 1 𝑘 superscript 𝑎 𝑑 𝑘 𝑛 superscript 𝑑 𝑛 2 superscript 1 𝑚 1 Euler-polynomial-E 𝑛 𝑚 𝑎 𝑑 Euler-polynomial-E 𝑛 𝑎 𝑑 {\displaystyle{\displaystyle\sum_{k=0}^{m-1}(-1)^{k}(a+dk)^{n}={\frac{d^{n}}{2% }\left((-1)^{m-1}E_{n}\left(m+\frac{a}{d}\right)+E_{n}\left(\frac{a}{d}\right)% \right)}}}
\sum_{k=0}^{m-1}(-1)^{k}(a+dk)^{n} = {\frac{d^{n}}{2}\left((-1)^{m-1}\EulerpolyE{n}@{m+\frac{a}{d}}+\EulerpolyE{n}@{\frac{a}{d}}\right)}		 

sum((- 1)^(k)*(a + d*k)^(n), k = 0..m - 1) = ((d)^(n))/(2)*((- 1)^(m - 1)* euler(n, m +(a)/(d))+ euler(n, (a)/(d)))

Sum[(- 1)^(k)*(a + d*k)^(n), {k, 0, m - 1}, GenerateConditions->None] == Divide[(d)^(n),2]*((- 1)^(m - 1)* EulerE[n, m +Divide[a,d]]+ EulerE[n, Divide[a,d]])
Failure Failure Successful [Tested: 300] Successful [Tested: 300]
24.4.E12 B n ⁡ ( x + h ) = ∑ k = 0 n ( n k ) ⁢ B k ⁡ ( x ) ⁢ h n - k Bernoulli-polynomial-B 𝑛 𝑥 ℎ superscript subscript 𝑘 0 𝑛 binomial 𝑛 𝑘 Bernoulli-polynomial-B 𝑘 𝑥 superscript ℎ 𝑛 𝑘 {\displaystyle{\displaystyle B_{n}\left(x+h\right)=\sum_{k=0}^{n}{n\choose k}B% _{k}\left(x\right)h^{n-k}}}
\BernoullipolyB{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\BernoullipolyB{k}@{x}h^{n-k}		 

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

BernoulliB[n, x + h] == Sum[Binomial[n,k]*BernoulliB[k, x]*(h)^(n - k), {k, 0, n}, GenerateConditions->None]
Failure Failure Successful [Tested: 90] Successful [Tested: 90]
24.4.E13 E n ⁡ ( x + h ) = ∑ k = 0 n ( n k ) ⁢ E k ⁡ ( x ) ⁢ h n - k Euler-polynomial-E 𝑛 𝑥 ℎ superscript subscript 𝑘 0 𝑛 binomial 𝑛 𝑘 Euler-polynomial-E 𝑘 𝑥 superscript ℎ 𝑛 𝑘 {\displaystyle{\displaystyle E_{n}\left(x+h\right)=\sum_{k=0}^{n}{n\choose k}E% _{k}\left(x\right)h^{n-k}}}
\EulerpolyE{n}@{x+h} = \sum_{k=0}^{n}{n\choose k}\EulerpolyE{k}@{x}h^{n-k}		 

euler(n, x + h) = sum(binomial(n,k)*euler(k, x)*(h)^(n - k), k = 0..n)

EulerE[n, x + h] == Sum[Binomial[n,k]*EulerE[k, x]*(h)^(n - k), {k, 0, n}, GenerateConditions->None]
Failure Failure Successful [Tested: 90] Successful [Tested: 90]
24.4.E14 E n - 1 ⁡ ( x ) = 2 n ⁢ ∑ k = 0 n ( n k ) ⁢ ( 1 - 2 k ) ⁢ B k ⁢ x n - k Euler-polynomial-E 𝑛 1 𝑥 2 𝑛 superscript subscript 𝑘 0 𝑛 binomial 𝑛 𝑘 1 superscript 2 𝑘 Bernoulli-number-B 𝑘 superscript 𝑥 𝑛 𝑘 {\displaystyle{\displaystyle E_{n-1}\left(x\right)=\frac{2}{n}\sum_{k=0}^{n}{n% \choose k}(1-2^{k})B_{k}x^{n-k}}}
\EulerpolyE{n-1}@{x} = \frac{2}{n}\sum_{k=0}^{n}{n\choose k}(1-2^{k})\BernoullinumberB{k}x^{n-k}		 

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

EulerE[n - 1, x] == Divide[2,n]*Sum[Binomial[n,k]*(1 - (2)^(k))*BernoulliB[k]*(x)^(n - k), {k, 0, n}, GenerateConditions->None]
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
24.4.E15 B 2 ⁢ n = 2 ⁢ n 2 2 ⁢ n ⁢ ( 2 2 ⁢ n - 1 ) ⁢ ∑ k = 0 n - 1 ( 2 ⁢ n - 1 2 ⁢ k ) ⁢ E 2 ⁢ k Bernoulli-number-B 2 𝑛 2 𝑛 superscript 2 2 𝑛 superscript 2 2 𝑛 1 superscript subscript 𝑘 0 𝑛 1 binomial 2 𝑛 1 2 𝑘 Euler-number-E 2 𝑘 {\displaystyle{\displaystyle B_{2n}=\frac{2n}{2^{2n}(2^{2n}-1)}\sum_{k=0}^{n-1% }{2n-1\choose 2k}E_{2k}}}
\BernoullinumberB{2n} = \frac{2n}{2^{2n}(2^{2n}-1)}\sum_{k=0}^{n-1}{2n-1\choose 2k}\EulernumberE{2k}		 

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

BernoulliB[2*n] == Divide[2*n,(2)^(2*n)*((2)^(2*n)- 1)]*Sum[Binomial[2*n - 1,2*k]*EulerE[2*k], {k, 0, n - 1}, GenerateConditions->None]
Missing Macro Error Failure - Successful [Tested: 3]
24.4.E16 E 2 ⁢ n = 1 2 ⁢ n + 1 - ∑ k = 1 n ( 2 ⁢ n 2 ⁢ k - 1 ) ⁢ 2 2 ⁢ k ⁢ ( 2 2 ⁢ k - 1 - 1 ) ⁢ B 2 ⁢ k k Euler-number-E 2 𝑛 1 2 𝑛 1 superscript subscript 𝑘 1 𝑛 binomial 2 𝑛 2 𝑘 1 superscript 2 2 𝑘 superscript 2 2 𝑘 1 1 Bernoulli-number-B 2 𝑘 𝑘 {\displaystyle{\displaystyle E_{2n}=\frac{1}{2n+1}-\sum_{k=1}^{n}{2n\choose 2k% -1}\frac{2^{2k}(2^{2k-1}-1)B_{2k}}{k}}}
\EulernumberE{2n} = \frac{1}{2n+1}-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k-1}-1)\BernoullinumberB{2k}}{k}		 

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

EulerE[2*n] == Divide[1,2*n + 1]- Sum[Binomial[2*n,2*k - 1]*Divide[(2)^(2*k)*((2)^(2*k - 1)- 1)*BernoulliB[2*k],k], {k, 1, n}, GenerateConditions->None]
Missing Macro Error Aborted - Skipped - Because timed out
24.4.E17 E 2 ⁢ n = 1 - ∑ k = 1 n ( 2 ⁢ n 2 ⁢ k - 1 ) ⁢ 2 2 ⁢ k ⁢ ( 2 2 ⁢ k - 1 ) ⁢ B 2 ⁢ k 2 ⁢ k Euler-number-E 2 𝑛 1 superscript subscript 𝑘 1 𝑛 binomial 2 𝑛 2 𝑘 1 superscript 2 2 𝑘 superscript 2 2 𝑘 1 Bernoulli-number-B 2 𝑘 2 𝑘 {\displaystyle{\displaystyle E_{2n}=1-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2% k}(2^{2k}-1)B_{2k}}{2k}}}
\EulernumberE{2n} = 1-\sum_{k=1}^{n}{2n\choose 2k-1}\frac{2^{2k}(2^{2k}-1)\BernoullinumberB{2k}}{2k}		 

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

EulerE[2*n] == 1 - Sum[Binomial[2*n,2*k - 1]*Divide[(2)^(2*k)*((2)^(2*k)- 1)*BernoulliB[2*k],2*k], {k, 1, n}, GenerateConditions->None]
Missing Macro Error Aborted - Skipped - Because timed out
24.4.E18 B n ⁡ ( m ⁢ x ) = m n - 1 ⁢ ∑ k = 0 m - 1 B n ⁡ ( x + k m ) Bernoulli-polynomial-B 𝑛 𝑚 𝑥 superscript 𝑚 𝑛 1 superscript subscript 𝑘 0 𝑚 1 Bernoulli-polynomial-B 𝑛 𝑥 𝑘 𝑚 {\displaystyle{\displaystyle B_{n}\left(mx\right)=m^{n-1}\sum_{k=0}^{m-1}B_{n}% \left(x+\frac{k}{m}\right)}}
\BernoullipolyB{n}@{mx} = m^{n-1}\sum_{k=0}^{m-1}\BernoullipolyB{n}@{x+\frac{k}{m}}		 

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

BernoulliB[n, m*x] == (m)^(n - 1)* Sum[BernoulliB[n, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]
Failure Successful Successful [Tested: 27] Successful [Tested: 27]
24.4.E19 E n ⁡ ( m ⁢ x ) = - 2 ⁢ m n n + 1 ⁢ ∑ k = 0 m - 1 ( - 1 ) k ⁢ B n + 1 ⁡ ( x + k m ) Euler-polynomial-E 𝑛 𝑚 𝑥 2 superscript 𝑚 𝑛 𝑛 1 superscript subscript 𝑘 0 𝑚 1 superscript 1 𝑘 Bernoulli-polynomial-B 𝑛 1 𝑥 𝑘 𝑚 {\displaystyle{\displaystyle E_{n}\left(mx\right)=-\frac{2m^{n}}{n+1}\sum_{k=0% }^{m-1}(-1)^{k}B_{n+1}\left(x+\frac{k}{m}\right)}}
\EulerpolyE{n}@{mx} = -\frac{2m^{n}}{n+1}\sum_{k=0}^{m-1}(-1)^{k}\BernoullipolyB{n+1}@{x+\frac{k}{m}}		 

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

EulerE[n, m*x] == -Divide[2*(m)^(n),n + 1]*Sum[(- 1)^(k)* BernoulliB[n + 1, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]
Failure Failure Successful [Tested: 9]
Failed [18 / 27]
Result: 1.9166666666666667
Test Values: {Rule[m, 1], Rule[n, 1], Rule[x, 1.5]}


Result: 1.25
Test Values: {Rule[m, 1], Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
24.4.E20 E n ⁡ ( m ⁢ x ) = m n ⁢ ∑ k = 0 m - 1 ( - 1 ) k ⁢ E n ⁡ ( x + k m ) Euler-polynomial-E 𝑛 𝑚 𝑥 superscript 𝑚 𝑛 superscript subscript 𝑘 0 𝑚 1 superscript 1 𝑘 Euler-polynomial-E 𝑛 𝑥 𝑘 𝑚 {\displaystyle{\displaystyle E_{n}\left(mx\right)=m^{n}\sum_{k=0}^{m-1}(-1)^{k% }E_{n}\left(x+\frac{k}{m}\right)}}
\EulerpolyE{n}@{mx} = m^{n}\sum_{k=0}^{m-1}(-1)^{k}\EulerpolyE{n}@{x+\frac{k}{m}}		 

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

EulerE[n, m*x] == (m)^(n)* Sum[(- 1)^(k)* EulerE[n, x +Divide[k,m]], {k, 0, m - 1}, GenerateConditions->None]
Failure Failure Successful [Tested: 9]
Failed [9 / 27]
Result: 3.5
Test Values: {Rule[m, 2], Rule[n, 1], Rule[x, 1.5]}


Result: 11.0
Test Values: {Rule[m, 2], Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
24.4.E21 B n ⁡ ( x ) = 2 n - 1 ⁢ ( B n ⁡ ( 1 2 ⁢ x ) + B n ⁡ ( 1 2 ⁢ x + 1 2 ) ) Bernoulli-polynomial-B 𝑛 𝑥 superscript 2 𝑛 1 Bernoulli-polynomial-B 𝑛 1 2 𝑥 Bernoulli-polynomial-B 𝑛 1 2 𝑥 1 2 {\displaystyle{\displaystyle B_{n}\left(x\right)=2^{n-1}\left(B_{n}\left(% \tfrac{1}{2}x\right)+B_{n}\left(\tfrac{1}{2}x+\tfrac{1}{2}\right)\right)}}
\BernoullipolyB{n}@{x} = 2^{n-1}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x}+\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}\right)		 

bernoulli(n, x) = (2)^(n - 1)*(bernoulli(n, (1)/(2)*x)+ bernoulli(n, (1)/(2)*x +(1)/(2)))

BernoulliB[n, x] == (2)^(n - 1)*(BernoulliB[n, Divide[1,2]*x]+ BernoulliB[n, Divide[1,2]*x +Divide[1,2]])
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
24.4.E22 E n - 1 ⁡ ( x ) = 2 n ⁢ ( B n ⁡ ( x ) - 2 n ⁢ B n ⁡ ( 1 2 ⁢ x ) ) Euler-polynomial-E 𝑛 1 𝑥 2 𝑛 Bernoulli-polynomial-B 𝑛 𝑥 superscript 2 𝑛 Bernoulli-polynomial-B 𝑛 1 2 𝑥 {\displaystyle{\displaystyle E_{n-1}\left(x\right)=\frac{2}{n}\left(B_{n}\left% (x\right)-2^{n}B_{n}\left(\tfrac{1}{2}x\right)\right)}}
\EulerpolyE{n-1}@{x} = \frac{2}{n}\left(\BernoullipolyB{n}@{x}-2^{n}\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)		 

euler(n - 1, x) = (2)/(n)*(bernoulli(n, x)- (2)^(n)* bernoulli(n, (1)/(2)*x))

EulerE[n - 1, x] == Divide[2,n]*(BernoulliB[n, x]- (2)^(n)* BernoulliB[n, Divide[1,2]*x])
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
24.4.E23 E n - 1 ⁡ ( x ) = 2 n n ⁢ ( B n ⁡ ( 1 2 ⁢ x + 1 2 ) - B n ⁡ ( 1 2 ⁢ x ) ) Euler-polynomial-E 𝑛 1 𝑥 superscript 2 𝑛 𝑛 Bernoulli-polynomial-B 𝑛 1 2 𝑥 1 2 Bernoulli-polynomial-B 𝑛 1 2 𝑥 {\displaystyle{\displaystyle E_{n-1}\left(x\right)=\frac{2^{n}}{n}\left(B_{n}% \left(\tfrac{1}{2}x+\tfrac{1}{2}\right)-B_{n}\left(\tfrac{1}{2}x\right)\right)}}
\EulerpolyE{n-1}@{x} = \frac{2^{n}}{n}\left(\BernoullipolyB{n}@{\tfrac{1}{2}x+\tfrac{1}{2}}-\BernoullipolyB{n}@{\tfrac{1}{2}x}\right)		 

euler(n - 1, x) = ((2)^(n))/(n)*(bernoulli(n, (1)/(2)*x +(1)/(2))- bernoulli(n, (1)/(2)*x))

EulerE[n - 1, x] == Divide[(2)^(n),n]*(BernoulliB[n, Divide[1,2]*x +Divide[1,2]]- BernoulliB[n, Divide[1,2]*x])
Failure Failure Successful [Tested: 9] Successful [Tested: 9]
24.4.E24 B n ⁡ ( m ⁢ x ) = m n ⁢ B n ⁡ ( x ) + n ⁢ ∑ k = 1 n ∑ j = 0 k - 1 ( - 1 ) j ⁢ ( n k ) ⁢ ( ∑ r = 1 m - 1 e 2 ⁢ π ⁢ i ⁢ ( k - j ) ⁢ r / m ( 1 - e 2 ⁢ π ⁢ i ⁢ r / m ) n ) ⁢ ( j + m ⁢ x ) n - 1 Bernoulli-polynomial-B 𝑛 𝑚 𝑥 superscript 𝑚 𝑛 Bernoulli-polynomial-B 𝑛 𝑥 𝑛 superscript subscript 𝑘 1 𝑛 superscript subscript 𝑗 0 𝑘 1 superscript 1 𝑗 binomial 𝑛 𝑘 superscript subscript 𝑟 1 𝑚 1 superscript 𝑒 2 𝜋 𝑖 𝑘 𝑗 𝑟 𝑚 superscript 1 superscript 𝑒 2 𝜋 𝑖 𝑟 𝑚 𝑛 superscript 𝑗 𝑚 𝑥 𝑛 1 {\displaystyle{\displaystyle B_{n}\left(mx\right)=m^{n}B_{n}\left(x\right)+n% \sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}% \frac{e^{2\pi i(k-j)r/m}}{(1-e^{2\pi ir/m})^{n}}\right)(j+mx)^{n-1}}}
\BernoullipolyB{n}@{mx} = m^{n}\BernoullipolyB{n}@{x}+n\sum_{k=1}^{n}\sum_{j=0}^{k-1}(-1)^{j}{n\choose k}\*\left(\sum_{r=1}^{m-1}\frac{e^{2\pi i(k-j)r/m}}{(1-e^{2\pi ir/m})^{n}}\right)(j+mx)^{n-1}		 

bernoulli(n, m*x) = (m)^(n)* bernoulli(n, x)+ n*sum(sum((- 1)^(j)*binomial(n,k)*(sum((exp(2*Pi*I*(k - j)*r/m))/((1 - exp(2*Pi*I*r/m))^(n)), r = 1..m - 1))*(j + m*x)^(n - 1), j = 0..k - 1), k = 1..n)

BernoulliB[n, m*x] == (m)^(n)* BernoulliB[n, x]+ n*Sum[Sum[(- 1)^(j)*Binomial[n,k]*(Sum[Divide[Exp[2*Pi*I*(k - j)*r/m],(1 - Exp[2*Pi*I*r/m])^(n)], {r, 1, m - 1}, GenerateConditions->None])*(j + m*x)^(n - 1), {j, 0, k - 1}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]
Aborted Failure Skipped - Because timed out
Failed [17 / 27]
Result: 1.0
Test Values: {Rule[m, 2], Rule[n, 1], Rule[x, 1.5]}


Result: 1.9999999999999991
Test Values: {Rule[m, 2], Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
24.4.E25 B n ⁡ ( 0 ) = ( - 1 ) n ⁢ B n ⁡ ( 1 ) Bernoulli-polynomial-B 𝑛 0 superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 1 {\displaystyle{\displaystyle B_{n}\left(0\right)=(-1)^{n}B_{n}\left(1\right)}}
\BernoullipolyB{n}@{0} = (-1)^{n}\BernoullipolyB{n}@{1}		 

bernoulli(n, 0) = (- 1)^(n)* bernoulli(n, 1)

BernoulliB[n, 0] == (- 1)^(n)* BernoulliB[n, 1]
Successful Failure - Successful [Tested: 3]
24.4.E25 ( - 1 ) n ⁢ B n ⁡ ( 1 ) = B n superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 1 Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle(-1)^{n}B_{n}\left(1\right)=B_{n}}}
(-1)^{n}\BernoullipolyB{n}@{1} = \BernoullinumberB{n}		 

(- 1)^(n)* bernoulli(n, 1) = bernoulli(n)

(- 1)^(n)* BernoulliB[n, 1] == BernoulliB[n]
Successful Failure - Successful [Tested: 3]
24.4.E26 E n ⁡ ( 0 ) = - E n ⁡ ( 1 ) Euler-polynomial-E 𝑛 0 Euler-polynomial-E 𝑛 1 {\displaystyle{\displaystyle E_{n}\left(0\right)=-E_{n}\left(1\right)}}
\EulerpolyE{n}@{0} = -\EulerpolyE{n}@{1}		 n > 0 𝑛 0 {\displaystyle{\displaystyle n>0}} 
euler(n, 0) = - euler(n, 1)

EulerE[n, 0] == - EulerE[n, 1]
Successful Successful Skip - symbolical successful subtest Successful [Tested: 3]
24.4.E26 - E n ⁡ ( 1 ) = - 2 n + 1 ⁢ ( 2 n + 1 - 1 ) ⁢ B n + 1 Euler-polynomial-E 𝑛 1 2 𝑛 1 superscript 2 𝑛 1 1 Bernoulli-number-B 𝑛 1 {\displaystyle{\displaystyle-E_{n}\left(1\right)=-\frac{2}{n+1}(2^{n+1}-1)B_{n% +1}}}
-\EulerpolyE{n}@{1} = -\frac{2}{n+1}(2^{n+1}-1)\BernoullinumberB{n+1}		 n > 0 𝑛 0 {\displaystyle{\displaystyle n>0}} 
- euler(n, 1) = -(2)/(n + 1)*((2)^(n + 1)- 1)*bernoulli(n + 1)

- EulerE[n, 1] == -Divide[2,n + 1]*((2)^(n + 1)- 1)*BernoulliB[n + 1]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.4.E27 B n ⁡ ( 1 2 ) = - ( 1 - 2 1 - n ) ⁢ B n Bernoulli-polynomial-B 𝑛 1 2 1 superscript 2 1 𝑛 Bernoulli-number-B 𝑛 {\displaystyle{\displaystyle B_{n}\left(\tfrac{1}{2}\right)=-(1-2^{1-n})B_{n}}}
\BernoullipolyB{n}@{\tfrac{1}{2}} = -(1-2^{1-n})\BernoullinumberB{n}		 

bernoulli(n, (1)/(2)) = -(1 - (2)^(1 - n))*bernoulli(n)

BernoulliB[n, Divide[1,2]] == -(1 - (2)^(1 - n))*BernoulliB[n]
Successful Successful - Successful [Tested: 3]
24.4.E28 E n ⁡ ( 1 2 ) = 2 - n ⁢ E n Euler-polynomial-E 𝑛 1 2 superscript 2 𝑛 Euler-number-E 𝑛 {\displaystyle{\displaystyle E_{n}\left(\tfrac{1}{2}\right)=2^{-n}E_{n}}}
\EulerpolyE{n}@{\tfrac{1}{2}} = 2^{-n}\EulernumberE{n}		 

euler(n, (1)/(2)) = (2)^(- n)* euler(n)

EulerE[n, Divide[1,2]] == (2)^(- n)* EulerE[n]
Missing Macro Error Successful - Successful [Tested: 3]
24.4.E29 B 2 ⁢ n ⁡ ( 1 3 ) = B 2 ⁢ n ⁡ ( 2 3 ) Bernoulli-polynomial-B 2 𝑛 1 3 Bernoulli-polynomial-B 2 𝑛 2 3 {\displaystyle{\displaystyle B_{2n}\left(\tfrac{1}{3}\right)=B_{2n}\left(% \tfrac{2}{3}\right)}}
\BernoullipolyB{2n}@{\tfrac{1}{3}} = \BernoullipolyB{2n}@{\tfrac{2}{3}}		 

bernoulli(2*n, (1)/(3)) = bernoulli(2*n, (2)/(3))

BernoulliB[2*n, Divide[1,3]] == BernoulliB[2*n, Divide[2,3]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.4.E29 B 2 ⁢ n ⁡ ( 2 3 ) = - 1 2 ⁢ ( 1 - 3 1 - 2 ⁢ n ) ⁢ B 2 ⁢ n Bernoulli-polynomial-B 2 𝑛 2 3 1 2 1 superscript 3 1 2 𝑛 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle B_{2n}\left(\tfrac{2}{3}\right)=-\tfrac{1}{2}(1-3% ^{1-2n})B_{2n}}}
\BernoullipolyB{2n}@{\tfrac{2}{3}} = -\tfrac{1}{2}(1-3^{1-2n})\BernoullinumberB{2n}		 

bernoulli(2*n, (2)/(3)) = -(1)/(2)*(1 - (3)^(1 - 2*n))*bernoulli(2*n)

BernoulliB[2*n, Divide[2,3]] == -Divide[1,2]*(1 - (3)^(1 - 2*n))*BernoulliB[2*n]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.4.E30 E 2 ⁢ n - 1 ⁡ ( 1 3 ) = - E 2 ⁢ n - 1 ⁡ ( 2 3 ) Euler-polynomial-E 2 𝑛 1 1 3 Euler-polynomial-E 2 𝑛 1 2 3 {\displaystyle{\displaystyle E_{2n-1}\left(\tfrac{1}{3}\right)=-E_{2n-1}\left(% \tfrac{2}{3}\right)}}
\EulerpolyE{2n-1}@{\tfrac{1}{3}} = -\EulerpolyE{2n-1}@{\tfrac{2}{3}}		 

euler(2*n - 1, (1)/(3)) = - euler(2*n - 1, (2)/(3))
 
EulerE[2*n - 1, Divide[1,3]] == - EulerE[2*n - 1, Divide[2,3]]
 Failure Failure Successful [Tested: 1] Successful [Tested: 3]
24.4.E30 - E 2 ⁢ n - 1 ⁡ ( 2 3 ) = - ( 1 - 3 1 - 2 ⁢ n ) ⁢ ( 2 2 ⁢ n - 1 ) 2 ⁢ n ⁢ B 2 ⁢ n Euler-polynomial-E 2 𝑛 1 2 3 1 superscript 3 1 2 𝑛 superscript 2 2 𝑛 1 2 𝑛 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle-E_{2n-1}\left(\tfrac{2}{3}\right)=-\frac{(1-3^{1-% 2n})(2^{2n}-1)}{2n}B_{2n}}}
-\EulerpolyE{2n-1}@{\tfrac{2}{3}} = -\frac{(1-3^{1-2n})(2^{2n}-1)}{2n}\BernoullinumberB{2n}		 

- euler(2*n - 1, (2)/(3)) = -((1 - (3)^(1 - 2*n))*((2)^(2*n)- 1))/(2*n)*bernoulli(2*n)
 
- EulerE[2*n - 1, Divide[2,3]] == -Divide[(1 - (3)^(1 - 2*n))*((2)^(2*n)- 1),2*n]*BernoulliB[2*n]
 Failure Failure Successful [Tested: 1] Successful [Tested: 3]
24.4.E31 B n ⁡ ( 1 4 ) = ( - 1 ) n ⁢ B n ⁡ ( 3 4 ) Bernoulli-polynomial-B 𝑛 1 4 superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 3 4 {\displaystyle{\displaystyle B_{n}\left(\tfrac{1}{4}\right)=(-1)^{n}B_{n}\left% (\tfrac{3}{4}\right)}}
\BernoullipolyB{n}@{\tfrac{1}{4}} = (-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}}		 

bernoulli(n, (1)/(4)) = (- 1)^(n)* bernoulli(n, (3)/(4))
 
BernoulliB[n, Divide[1,4]] == (- 1)^(n)* BernoulliB[n, Divide[3,4]]
 Failure Successful Successful [Tested: 1] Successful [Tested: 1]
24.4.E31 ( - 1 ) n ⁢ B n ⁡ ( 3 4 ) = - 1 - 2 1 - n 2 n ⁢ B n - n 4 n ⁢ E n - 1 superscript 1 𝑛 Bernoulli-polynomial-B 𝑛 3 4 1 superscript 2 1 𝑛 superscript 2 𝑛 Bernoulli-number-B 𝑛 𝑛 superscript 4 𝑛 Euler-number-E 𝑛 1 {\displaystyle{\displaystyle(-1)^{n}B_{n}\left(\tfrac{3}{4}\right)=-\frac{1-2^% {1-n}}{2^{n}}B_{n}-\frac{n}{4^{n}}E_{n-1}}}
(-1)^{n}\BernoullipolyB{n}@{\tfrac{3}{4}} = -\frac{1-2^{1-n}}{2^{n}}\BernoullinumberB{n}-\frac{n}{4^{n}}\EulernumberE{n-1}		 

(- 1)^(n)* bernoulli(n, (3)/(4)) = -(1 - (2)^(1 - n))/((2)^(n))*bernoulli(n)-(n)/((4)^(n))*euler(n - 1)
 
(- 1)^(n)* BernoulliB[n, Divide[3,4]] == -Divide[1 - (2)^(1 - n),(2)^(n)]*BernoulliB[n]-Divide[n,(4)^(n)]*EulerE[n - 1]
 Missing Macro Error Failure Skip - symbolical successful subtest Successful [Tested: 3]
24.4.E32 B 2 ⁢ n ⁡ ( 1 6 ) = B 2 ⁢ n ⁡ ( 5 6 ) Bernoulli-polynomial-B 2 𝑛 1 6 Bernoulli-polynomial-B 2 𝑛 5 6 {\displaystyle{\displaystyle B_{2n}\left(\tfrac{1}{6}\right)=B_{2n}\left(% \tfrac{5}{6}\right)}}
\BernoullipolyB{2n}@{\tfrac{1}{6}} = \BernoullipolyB{2n}@{\tfrac{5}{6}}		 

bernoulli(2*n, (1)/(6)) = bernoulli(2*n, (5)/(6))
 
BernoulliB[2*n, Divide[1,6]] == BernoulliB[2*n, Divide[5,6]]
 Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.4.E32 B 2 ⁢ n ⁡ ( 5 6 ) = 1 2 ⁢ ( 1 - 2 1 - 2 ⁢ n ) ⁢ ( 1 - 3 1 - 2 ⁢ n ) ⁢ B 2 ⁢ n Bernoulli-polynomial-B 2 𝑛 5 6 1 2 1 superscript 2 1 2 𝑛 1 superscript 3 1 2 𝑛 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle B_{2n}\left(\tfrac{5}{6}\right)=\tfrac{1}{2}(1-2^% {1-2n})(1-3^{1-2n})B_{2n}}}
\BernoullipolyB{2n}@{\tfrac{5}{6}} = \tfrac{1}{2}(1-2^{1-2n})(1-3^{1-2n})\BernoullinumberB{2n}		 

bernoulli(2*n, (5)/(6)) = (1)/(2)*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))*bernoulli(2*n)
 
BernoulliB[2*n, Divide[5,6]] == Divide[1,2]*(1 - (2)^(1 - 2*n))*(1 - (3)^(1 - 2*n))*BernoulliB[2*n]
 Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.4.E33 E 2 ⁢ n ⁡ ( 1 6 ) = E 2 ⁢ n ⁡ ( 5 6 ) Euler-polynomial-E 2 𝑛 1 6 Euler-polynomial-E 2 𝑛 5 6 {\displaystyle{\displaystyle E_{2n}\left(\tfrac{1}{6}\right)=E_{2n}\left(% \tfrac{5}{6}\right)}}
\EulerpolyE{2n}@{\tfrac{1}{6}} = \EulerpolyE{2n}@{\tfrac{5}{6}}		 

euler(2*n, (1)/(6)) = euler(2*n, (5)/(6))
 
EulerE[2*n, Divide[1,6]] == EulerE[2*n, Divide[5,6]]
 Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.4.E33 E 2 ⁢ n ⁡ ( 5 6 ) = 1 + 3 - 2 ⁢ n 2 2 ⁢ n + 1 ⁢ E 2 ⁢ n Euler-polynomial-E 2 𝑛 5 6 1 superscript 3 2 𝑛 superscript 2 2 𝑛 1 Euler-number-E 2 𝑛 {\displaystyle{\displaystyle E_{2n}\left(\tfrac{5}{6}\right)=\frac{1+3^{-2n}}{% 2^{2n+1}}E_{2n}}}
\EulerpolyE{2n}@{\tfrac{5}{6}} = \frac{1+3^{-2n}}{2^{2n+1}}\EulernumberE{2n}		 

euler(2*n, (5)/(6)) = (1 + (3)^(- 2*n))/((2)^(2*n + 1))*euler(2*n)
 
EulerE[2*n, Divide[5,6]] == Divide[1 + (3)^(- 2*n),(2)^(2*n + 1)]*EulerE[2*n]
 Missing Macro Error Failure Skip - symbolical successful subtest Successful [Tested: 3]
24.4.E34 d d x ⁡ B n ⁡ ( x ) = n ⁢ B n - 1 ⁡ ( x ) derivative 𝑥 Bernoulli-polynomial-B 𝑛 𝑥 𝑛 Bernoulli-polynomial-B 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}B_{n}\left(x\right)=% nB_{n-1}\left(x\right)}}
\deriv{}{x}\BernoullipolyB{n}@{x} = n\BernoullipolyB{n-1}@{x}		 

diff(bernoulli(n, x), x) = n*bernoulli(n - 1, x)
 
D[BernoulliB[n, x], x] == n*BernoulliB[n - 1, x]
 Successful Successful - Successful [Tested: 3]
24.4.E35 d d x ⁡ E n ⁡ ( x ) = n ⁢ E n - 1 ⁡ ( x ) derivative 𝑥 Euler-polynomial-E 𝑛 𝑥 𝑛 Euler-polynomial-E 𝑛 1 𝑥 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}E_{n}\left(x\right)=% nE_{n-1}\left(x\right)}}
\deriv{}{x}\EulerpolyE{n}@{x} = n\EulerpolyE{n-1}@{x}		 

diff(euler(n, x), x) = n*euler(n - 1, x)
 
D[EulerE[n, x], x] == n*EulerE[n - 1, x]
 Successful Successful - Successful [Tested: 3]
24.4.E37 B n ⁡ ( x + h ) = ( B ⁢ ( x ) + h ) n Bernoulli-polynomial-B 𝑛 𝑥 ℎ superscript 𝐵 𝑥 ℎ 𝑛 {\displaystyle{\displaystyle B_{n}\left(x+h\right)=(B(x)+h)^{n}}}
\BernoullipolyB{n}@{x+h} = (B(x)+h)^{n}		 

bernoulli(n, x + h) = (B(x)+ h)^(n)
 
BernoulliB[n, x + h] == (B[x]+ h)^(n)
 Failure Failure
Failed [300 / 300]
Result: -.299038106-.7500000000*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1}

Result: .23717473e-1-3.546633371*I
Test Values: {B = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.299038105676658, -0.7499999999999998]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1], Rule[x, 1.5]}

Result: Complex[0.023717474235543268, -3.546633369868303]
Test Values: {Rule[B, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
24.4.E38 P ⁢ ( E ⁢ ( x ) + 1 ) + P ⁢ ( E ⁢ ( x ) ) = 2 ⁢ P ⁢ ( x ) 𝑃 𝐸 𝑥 1 𝑃 𝐸 𝑥 2 𝑃 𝑥 {\displaystyle{\displaystyle P(E(x)+1)+P(E(x))=2P(x)}}
P(E(x)+1)+P(E(x)) = 2P(x)		 

P(E(x)+ 1)+ P(E(x)) = 2*P(x)
 
P[E[x]+ 1]+ P[E[x]] == 2*P[x]
 Skipped - no semantic math Skipped - no semantic math - -
24.4.E39 E n ⁡ ( x + h ) = ( E ⁢ ( x ) + h ) n Euler-polynomial-E 𝑛 𝑥 ℎ superscript 𝐸 𝑥 ℎ 𝑛 {\displaystyle{\displaystyle E_{n}\left(x+h\right)=(E(x)+h)^{n}}}
\EulerpolyE{n}@{x+h} = (E(x)+h)^{n}		 

euler(n, x + h) = (E(x)+ h)^(n)
 
EulerE[n, x + h] == (E[x]+ h)^(n)
 Failure Failure
Failed [300 / 300]
Result: -.299038106-.7500000000*I
Test Values: {E = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 1}

Result: -.142949194-3.546633371*I
Test Values: {E = 1/2*3^(1/2)+1/2*I, h = 1/2*3^(1/2)+1/2*I, x = 3/2, n = 2}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.299038105676658, -0.7499999999999998]
Test Values: {Rule[E, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 1], Rule[x, 1.5]}

Result: Complex[-0.14294919243112325, -3.546633369868303]
Test Values: {Rule[E, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[n, 2], Rule[x, 1.5]}

... skip entries to safe data
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