Bernoulli and Euler Polynomials - 25.2 Definition and Expansions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
25.2.E1 ΞΆ ⁑ ( s ) = βˆ‘ n = 1 ∞ 1 n s Riemann-zeta 𝑠 superscript subscript 𝑛 1 1 superscript 𝑛 𝑠 {\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{n=1}^{\infty}\frac{1}{n^% {s}}}}
\Riemannzeta@{s} = \sum_{n=1}^{\infty}\frac{1}{n^{s}}		 

Zeta(s) = sum((1)/((n)^(s)), n = 1..infinity)

Zeta[s] == Sum[Divide[1,(n)^(s)], {n, 1, Infinity}, GenerateConditions->None]
Failure Successful
Failed [4 / 6]
Result: Float(-infinity)
Test Values: {s = -3/2}


Result: Float(-infinity)
Test Values: {s = -1/2}

... skip entries to safe data
 Successful [Tested: 6]
25.2.E2 ΞΆ ⁑ ( s ) = 1 1 - 2 - s ⁒ βˆ‘ n = 0 ∞ 1 ( 2 ⁒ n + 1 ) s Riemann-zeta 𝑠 1 1 superscript 2 𝑠 superscript subscript 𝑛 0 1 superscript 2 𝑛 1 𝑠 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{1-2^{-s}}\sum_{n=0}^{% \infty}\frac{1}{(2n+1)^{s}}}}
\Riemannzeta@{s} = \frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}}		 β„œ ⁑ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}} 
Zeta(s) = (1)/(1 - (2)^(- s))*sum((1)/((2*n + 1)^(s)), n = 0..infinity)

Zeta[s] == Divide[1,1 - (2)^(- s)]*Sum[Divide[1,(2*n + 1)^(s)], {n, 0, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 2]
25.2.E3 ΞΆ ⁑ ( s ) = 1 1 - 2 1 - s ⁒ βˆ‘ n = 1 ∞ ( - 1 ) n - 1 n s Riemann-zeta 𝑠 1 1 superscript 2 1 𝑠 superscript subscript 𝑛 1 superscript 1 𝑛 1 superscript 𝑛 𝑠 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{1-2^{1-s}}\sum_{n=1}^% {\infty}\frac{(-1)^{n-1}}{n^{s}}}}
\Riemannzeta@{s} = \frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}		 β„œ ⁑ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}} 
Zeta(s) = (1)/(1 - (2)^(1 - s))*sum(((- 1)^(n - 1))/((n)^(s)), n = 1..infinity)

Zeta[s] == Divide[1,1 - (2)^(1 - s)]*Sum[Divide[(- 1)^(n - 1),(n)^(s)], {n, 1, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
25.2.E4 ΞΆ ⁑ ( s ) = 1 s - 1 + βˆ‘ n = 0 ∞ ( - 1 ) n n ! ⁒ Ξ³ n ⁒ ( s - 1 ) n Riemann-zeta 𝑠 1 𝑠 1 superscript subscript 𝑛 0 superscript 1 𝑛 𝑛 Stieltjes-constants 𝑛 superscript 𝑠 1 𝑛 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{% \infty}\frac{(-1)^{n}}{n!}\gamma_{n}(s-1)^{n}}}
\Riemannzeta@{s} = \frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\StieltjesConstants{n}(s-1)^{n}		 

Zeta(s) = (1)/(s - 1)+ sum(((- 1)^(n))/(factorial(n))*gamma(n)*(s - 1)^(n), n = 0..infinity)

Error
Failure Missing Macro Error Successful [Tested: 6] -
25.2.E5 Ξ³ n = lim m β†’ ∞ ⁑ ( βˆ‘ k = 1 m ( ln ⁑ k ) n k - ( ln ⁑ m ) n + 1 n + 1 ) Stieltjes-constants 𝑛 subscript β†’ π‘š superscript subscript π‘˜ 1 π‘š superscript π‘˜ 𝑛 π‘˜ superscript π‘š 𝑛 1 𝑛 1 {\displaystyle{\displaystyle\gamma_{n}=\lim_{m\to\infty}\left(\sum_{k=1}^{m}% \frac{(\ln k)^{n}}{k}-\frac{(\ln m)^{n+1}}{n+1}\right)}}
\StieltjesConstants{n} = \lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln@@{k})^{n}}{k}-\frac{(\ln@@{m})^{n+1}}{n+1}\right)		 

gamma(n) = limit(sum(((ln(k))^(n))/(k), k = 1..m)-((ln(m))^(n + 1))/(n + 1), m = infinity)

Error
Successful Missing Macro Error - -
25.2.E6 ΞΆ β€² ⁑ ( s ) = - βˆ‘ n = 2 ∞ ( ln ⁑ n ) ⁒ n - s diffop Riemann-zeta 1 𝑠 superscript subscript 𝑛 2 𝑛 superscript 𝑛 𝑠 {\displaystyle{\displaystyle\zeta'\left(s\right)=-\sum_{n=2}^{\infty}(\ln n)n^% {-s}}}
\Riemannzeta'@{s} = -\sum_{n=2}^{\infty}(\ln@@{n})n^{-s}		 β„œ ⁑ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}} 
diff( Zeta(s), s$(1) ) = - sum((ln(n))*(n)^(- s), n = 2..infinity)

D[Zeta[s], {s, 1}] == - Sum[(Log[n])*(n)^(- s), {n, 2, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 2]
25.2.E7 ΞΆ ( k ) ⁑ ( s ) = ( - 1 ) k ⁒ βˆ‘ n = 2 ∞ ( ln ⁑ n ) k ⁒ n - s Riemann-zeta π‘˜ 𝑠 superscript 1 π‘˜ superscript subscript 𝑛 2 superscript 𝑛 π‘˜ superscript 𝑛 𝑠 {\displaystyle{\displaystyle{\zeta^{(k)}}\left(s\right)=(-1)^{k}\sum_{n=2}^{% \infty}(\ln n)^{k}n^{-s}}}
\Riemannzeta^{(k)}@{s} = (-1)^{k}\sum_{n=2}^{\infty}(\ln@@{n})^{k}n^{-s}		 β„œ ⁑ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}} 
diff( Zeta(s), s$(k) ) = (- 1)^(k)* sum((ln(n))^(k)* (n)^(- s), n = 2..infinity)

D[Zeta[s], {s, k}] == (- 1)^(k)* Sum[(Log[n])^(k)* (n)^(- s), {n, 2, Infinity}, GenerateConditions->None]
Failure Failure Successful [Tested: 2] Successful [Tested: 2]
25.2.E8 ΞΆ ⁑ ( s ) = βˆ‘ k = 1 N 1 k s + N 1 - s s - 1 - s ⁒ ∫ N ∞ x - ⌊ x βŒ‹ x s + 1 ⁒ d x Riemann-zeta 𝑠 superscript subscript π‘˜ 1 𝑁 1 superscript π‘˜ 𝑠 superscript 𝑁 1 𝑠 𝑠 1 𝑠 superscript subscript 𝑁 π‘₯ π‘₯ superscript π‘₯ 𝑠 1 π‘₯ {\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+% \frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{% s+1}}\mathrm{d}x}}
\Riemannzeta@{s} = \sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\floor{x}}{x^{s+1}}\diff{x}		 β„œ ⁑ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}} 
Zeta(s) = sum((1)/((k)^(s)), k = 1..N)+((N)^(1 - s))/(s - 1)- s*int((x - floor(x))/((x)^(s + 1)), x = N..infinity)

Zeta[s] == Sum[Divide[1,(k)^(s)], {k, 1, N}, GenerateConditions->None]+Divide[(N)^(1 - s),s - 1]- s*Integrate[Divide[x - Floor[x],(x)^(s + 1)], {x, N, Infinity}, GenerateConditions->None]
Failure Aborted
Failed [3 / 3]
Result: .2180864797
Test Values: {s = 3/2, N = 3}


Result: Float(infinity)
Test Values: {s = 1/2, N = 3}

... skip entries to safe data
 Skipped - Because timed out
25.2.E11 ΞΆ ⁑ ( s ) = ∏ p ( 1 - p - s ) - 1 Riemann-zeta 𝑠 subscript product 𝑝 superscript 1 superscript 𝑝 𝑠 1 {\displaystyle{\displaystyle\zeta\left(s\right)=\prod_{p}(1-p^{-s})^{-1}}}
\Riemannzeta@{s} = \prod_{p}(1-p^{-s})^{-1}		 β„œ ⁑ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}} 
Zeta(s) = product((1 - (p)^(- s))^(- 1), p = - infinity..infinity)

Zeta[s] == Product[(1 - (p)^(- s))^(- 1), {p, - Infinity, Infinity}, GenerateConditions->None]
Failure Failure Error
Failed [2 / 2]
Result: Plus[2.612375348685488, Times[-1.0, NProduct[Power[Plus[1, Times[-1, Power[p, -1.5]]], -1]
Test Values: {p, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 1.5]}


Result: Plus[1.6449340668482262, Times[-1.0, NProduct[Power[Plus[1, Times[-1, Power[p, -2]]], -1]
Test Values: {p, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 2]}

25.2.E12 ΞΆ ⁑ ( s ) = ( 2 ⁒ Ο€ ) s ⁒ e - s - ( Ξ³ ⁒ s / 2 ) 2 ⁒ ( s - 1 ) ⁒ Ξ“ ⁑ ( 1 2 ⁒ s + 1 ) ⁒ ∏ ρ ( 1 - s ρ ) ⁒ e s / ρ Riemann-zeta 𝑠 superscript 2 πœ‹ 𝑠 superscript 𝑒 𝑠 𝑠 2 2 𝑠 1 Euler-Gamma 1 2 𝑠 1 subscript product 𝜌 1 𝑠 𝜌 superscript 𝑒 𝑠 𝜌 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{(2\pi)^{s}e^{-s-(\gamma s% /2)}}{2(s-1)\Gamma\left(\tfrac{1}{2}s+1\right)}\prod_{\rho}\left(1-\frac{s}{% \rho}\right)e^{s/\rho}}}
\Riemannzeta@{s} = \frac{(2\pi)^{s}e^{-s-(\EulerConstant s/2)}}{2(s-1)\EulerGamma@{\tfrac{1}{2}s+1}}\prod_{\rho}\left(1-\frac{s}{\rho}\right)e^{s/\rho}		 β„œ ⁑ ( 1 2 ⁒ s + 1 ) > 0 1 2 𝑠 1 0 {\displaystyle{\displaystyle\Re(\tfrac{1}{2}s+1)>0}} 
Zeta(s) = ((2*Pi)^(s)* exp(- s -(gamma*s/2)))/(2*(s - 1)*GAMMA((1)/(2)*s + 1))*product((1 -(s)/(rho))*exp(s/rho), rho = - infinity..infinity)

Zeta[s] == Divide[(2*Pi)^(s)* Exp[- s -(EulerGamma*s/2)],2*(s - 1)*Gamma[Divide[1,2]*s + 1]]*Product[(1 -Divide[s,\[Rho]])*Exp[s/\[Rho]], {\[Rho], - Infinity, Infinity}, GenerateConditions->None]
Failure Failure Error
Failed [5 / 5]
Result: Plus[Complex[-0.02548520188983307, 3.121037092000815*^-18], Times[0.02420092827533985, NProduct[Times[Power[E, Times[-1.5, Power[ρ, -1]]], Plus[1, Times[1.5, Power[ρ, -1]]]]
Test Values: {ρ, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, -1.5]}


Result: Plus[2.612375348685488, Times[-2.4801151038890965, NProduct[Times[Power[E, Times[1.5, Power[ρ, -1]]], Plus[1, Times[-1.5, Power[ρ, -1]]]]
Test Values: {ρ, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 1.5]}

... skip entries to safe data
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