Bernoulli and Euler Polynomials - 24.2 Definitions and Generating Functions

From testwiki
Revision as of 12:01, 28 June 2021 by Admin (talk | contribs) (Admin moved page Main Page to Verifying DLMF with Maple and Mathematica)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
24.2.E1 t e t - 1 = ∑ n = 0 ∞ B n ⁢ t n n ! 𝑡 superscript 𝑒 𝑡 1 superscript subscript 𝑛 0 Bernoulli-number-B 𝑛 superscript 𝑡 𝑛 𝑛 {\displaystyle{\displaystyle\frac{t}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}\frac{t^% {n}}{n!}}}
\frac{t}{e^{t}-1} = \sum_{n=0}^{\infty}\BernoullinumberB{n}\frac{t^{n}}{n!}		 | t | < 2 ⁢ π 𝑡 2 𝜋 {\displaystyle{\displaystyle|t|<2\pi}} 
(t)/(exp(t)- 1) = sum(bernoulli(n)*((t)^(n))/(factorial(n)), n = 0..infinity)

Divide[t,Exp[t]- 1] == Sum[BernoulliB[n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 6] Successful [Tested: 6]
24.2#Ex1 B 2 ⁢ n + 1 = 0 Bernoulli-number-B 2 𝑛 1 0 {\displaystyle{\displaystyle B_{2n+1}=0}}
\BernoullinumberB{2n+1} = 0		 

bernoulli(2*n + 1) = 0

BernoulliB[2*n + 1] == 0
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
24.2#Ex2 ( - 1 ) n + 1 ⁢ B 2 ⁢ n > 0 superscript 1 𝑛 1 Bernoulli-number-B 2 𝑛 0 {\displaystyle{\displaystyle(-1)^{n+1}B_{2n}>0}}
(-1)^{n+1}\BernoullinumberB{2n} > 0		 

(- 1)^(n + 1)* bernoulli(2*n) > 0

(- 1)^(n + 1)* BernoulliB[2*n] > 0
Failure Failure Successful [Tested: 1] Successful [Tested: 3]
24.2.E3 t ⁢ e x ⁢ t e t - 1 = ∑ n = 0 ∞ B n ⁡ ( x ) ⁢ t n n ! 𝑡 superscript 𝑒 𝑥 𝑡 superscript 𝑒 𝑡 1 superscript subscript 𝑛 0 Bernoulli-polynomial-B 𝑛 𝑥 superscript 𝑡 𝑛 𝑛 {\displaystyle{\displaystyle\frac{te^{xt}}{e^{t}-1}=\sum_{n=0}^{\infty}B_{n}% \left(x\right)\frac{t^{n}}{n!}}}
\frac{te^{xt}}{e^{t}-1} = \sum_{n=0}^{\infty}\BernoullipolyB{n}@{x}\frac{t^{n}}{n!}		 | t | < 2 ⁢ π 𝑡 2 𝜋 {\displaystyle{\displaystyle|t|<2\pi}} 
(t*exp(x*t))/(exp(t)- 1) = sum(bernoulli(n, x)*((t)^(n))/(factorial(n)), n = 0..infinity)

Divide[t*Exp[x*t],Exp[t]- 1] == Sum[BernoulliB[n, x]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 18] Successful [Tested: 18]
24.2.E4 B n = B n ⁡ ( 0 ) Bernoulli-number-B 𝑛 Bernoulli-polynomial-B 𝑛 0 {\displaystyle{\displaystyle B_{n}=B_{n}\left(0\right)}}
\BernoullinumberB{n} = \BernoullipolyB{n}@{0}		 

bernoulli(n) = bernoulli(n, 0)

BernoulliB[n] == BernoulliB[n, 0]
Successful Successful - Successful [Tested: 3]
24.2.E5 B n ⁡ ( x ) = ∑ k = 0 n ( n k ) ⁢ B k ⁢ x n - k Bernoulli-polynomial-B 𝑛 𝑥 superscript subscript 𝑘 0 𝑛 binomial 𝑛 𝑘 Bernoulli-number-B 𝑘 superscript 𝑥 𝑛 𝑘 {\displaystyle{\displaystyle B_{n}\left(x\right)=\sum_{k=0}^{n}{n\choose k}B_{% k}x^{n-k}}}
\BernoullipolyB{n}@{x} = \sum_{k=0}^{n}{n\choose k}\BernoullinumberB{k}x^{n-k}		 

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

BernoulliB[n, x] == Sum[Binomial[n,k]*BernoulliB[k]*(x)^(n - k), {k, 0, n}, GenerateConditions->None]
Failure Successful Successful [Tested: 9] Successful [Tested: 9]
24.2.E6 2 ⁢ e t e 2 ⁢ t + 1 = ∑ n = 0 ∞ E n ⁢ t n n ! 2 superscript 𝑒 𝑡 superscript 𝑒 2 𝑡 1 superscript subscript 𝑛 0 Euler-number-E 𝑛 superscript 𝑡 𝑛 𝑛 {\displaystyle{\displaystyle\frac{2e^{t}}{e^{2t}+1}=\sum_{n=0}^{\infty}E_{n}% \frac{t^{n}}{n!}}}
\frac{2e^{t}}{e^{2t}+1} = \sum_{n=0}^{\infty}\EulernumberE{n}\frac{t^{n}}{n!}		 | t | < 1 2 ⁢ π 𝑡 1 2 𝜋 {\displaystyle{\displaystyle|t|<\tfrac{1}{2}\pi}} 
(2*exp(t))/(exp(2*t)+ 1) = sum(euler(n)*((t)^(n))/(factorial(n)), n = 0..infinity)

Divide[2*Exp[t],Exp[2*t]+ 1] == Sum[EulerE[n]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]
Missing Macro Error Successful - Successful [Tested: 4]
24.2#Ex3 E 2 ⁢ n + 1 = 0 Euler-number-E 2 𝑛 1 0 {\displaystyle{\displaystyle E_{2n+1}=0}}
\EulernumberE{2n+1} = 0		 

euler(2*n + 1) = 0

EulerE[2*n + 1] == 0
Missing Macro Error Failure - Successful [Tested: 3]
24.2#Ex4 ( - 1 ) n ⁢ E 2 ⁢ n > 0 superscript 1 𝑛 Euler-number-E 2 𝑛 0 {\displaystyle{\displaystyle(-1)^{n}E_{2n}>0}}
(-1)^{n}\EulernumberE{2n} > 0		 

(- 1)^(n)* euler(2*n) > 0

(- 1)^(n)* EulerE[2*n] > 0
Missing Macro Error Failure - Successful [Tested: 3]
24.2.E8 2 ⁢ e x ⁢ t e t + 1 = ∑ n = 0 ∞ E n ⁡ ( x ) ⁢ t n n ! 2 superscript 𝑒 𝑥 𝑡 superscript 𝑒 𝑡 1 superscript subscript 𝑛 0 Euler-polynomial-E 𝑛 𝑥 superscript 𝑡 𝑛 𝑛 {\displaystyle{\displaystyle\frac{2e^{xt}}{e^{t}+1}=\sum_{n=0}^{\infty}E_{n}% \left(x\right)\frac{t^{n}}{n!}}}
\frac{2e^{xt}}{e^{t}+1} = \sum_{n=0}^{\infty}\EulerpolyE{n}@{x}\frac{t^{n}}{n!}		 | t | < π 𝑡 𝜋 {\displaystyle{\displaystyle|t|<\pi}} 
(2*exp(x*t))/(exp(t)+ 1) = sum(euler(n, x)*((t)^(n))/(factorial(n)), n = 0..infinity)

Divide[2*Exp[x*t],Exp[t]+ 1] == Sum[EulerE[n, x]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]
Failure Successful Error Successful [Tested: 18]
24.2.E9 E n = 2 n ⁢ E n ⁡ ( 1 2 ) Euler-number-E 𝑛 superscript 2 𝑛 Euler-polynomial-E 𝑛 1 2 {\displaystyle{\displaystyle E_{n}=2^{n}E_{n}\left(\tfrac{1}{2}\right)}}
\EulernumberE{n} = 2^{n}\EulerpolyE{n}@{\tfrac{1}{2}}		 

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

EulerE[n] == (2)^(n)* EulerE[n, Divide[1,2]]
Missing Macro Error Successful - Successful [Tested: 3]
24.2.E10 E n ⁡ ( x ) = ∑ k = 0 n ( n k ) ⁢ E k 2 k ⁢ ( x - 1 2 ) n - k Euler-polynomial-E 𝑛 𝑥 superscript subscript 𝑘 0 𝑛 binomial 𝑛 𝑘 Euler-number-E 𝑘 superscript 2 𝑘 superscript 𝑥 1 2 𝑛 𝑘 {\displaystyle{\displaystyle E_{n}\left(x\right)=\sum_{k=0}^{n}{n\choose k}% \frac{E_{k}}{2^{k}}(x-\tfrac{1}{2})^{n-k}}}
\EulerpolyE{n}@{x} = \sum_{k=0}^{n}{n\choose k}\frac{\EulernumberE{k}}{2^{k}}(x-\tfrac{1}{2})^{n-k}		 

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

EulerE[n, x] == Sum[Binomial[n,k]*Divide[EulerE[k],(2)^(k)]*(x -Divide[1,2])^(n - k), {k, 0, n}, GenerateConditions->None]
Missing Macro Error Failure -
Failed [3 / 9]
Result: Indeterminate
Test Values: {Rule[n, 1], Rule[x, 0.5]}


Result: Indeterminate
Test Values: {Rule[n, 2], Rule[x, 0.5]}

... skip entries to safe data
Retrieved from "https://lct.wmflabs.org/w/index.php?title=24.2&oldid=58434"