Bernoulli and Euler Polynomials - 24.16 Generalizations

From testwiki
Jump to navigation Jump to search


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
24.16.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{t}{\ln@{1+t}} = \sum_{n=0}^{\infty}b_{n}t^{n}}
\frac{t}{\ln@{1+t}} = \sum_{n=0}^{\infty}b_{n}t^{n}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |t| < 1} 
(t)/(ln(1 + t)) = sum(b[n]*(t)^(n), n = 0..infinity)

Divide[t,Log[1 + t]] == Sum[Subscript[b, n]*(t)^(n), {n, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [20 / 20]
Result: .1439972511-.3333333333*I
Test Values: {t = -1/2, b[n] = 1/2*3^(1/2)+1/2*I}


Result: 1.054680854-.5773502693*I
Test Values: {t = -1/2, b[n] = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [20 / 20]
Result: Complex[0.14399725125485596, -0.33333333333333326]
Test Values: {Rule[t, -0.5], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.054680853777815, -0.5773502691896257]
Test Values: {Rule[t, -0.5], Rule[Subscript[b, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
24.16.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{t}{(1+\lambda t)^{\ifrac{1}{\lambda}}-1} = \sum_{n=0}^{\infty}\beta_{n}(\lambda)\frac{t^{n}}{n!}}
\frac{t}{(1+\lambda t)^{\ifrac{1}{\lambda}}-1} = \sum_{n=0}^{\infty}\beta_{n}(\lambda)\frac{t^{n}}{n!}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(t)/((1 + lambda*t)^((1)/(lambda))- 1) = sum(beta[n](lambda)*((t)^(n))/(factorial(n)), n = 0..infinity)
 
Divide[t,(1 + \[Lambda]*t)^(Divide[1,\[Lambda]])- 1] == Sum[Subscript[\[Beta], n][\[Lambda]]*Divide[(t)^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
24.16.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \beta_{n}(\lambda) = n!b_{n}\lambda^{n}+\sum_{k=1}^{\floor{\ifrac{n}{2}}}\frac{n}{2k}\BernoullinumberB{2k}\Stirlingnumbers@{n-1}{2k-1}\lambda^{n-2k}}
\beta_{n}(\lambda) = n!b_{n}\lambda^{n}+\sum_{k=1}^{\floor{\ifrac{n}{2}}}\frac{n}{2k}\BernoullinumberB{2k}\Stirlingnumbers@{n-1}{2k-1}\lambda^{n-2k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
beta[n](lambda) = factorial(n)*b[n]*(lambda)^(n)+ sum((n)/(2*k)*bernoulli(2*k)*Stirling1(n - 1, 2*k - 1)*(lambda)^(n - 2*k), k = 1..floor((n)/(2)))

Subscript[\[Beta], n][\[Lambda]] == (n)!*Subscript[b, n]*\[Lambda]^(n)+ Sum[Divide[n,2*k]*BernoulliB[2*k]*StirlingS1[n - 1, 2*k - 1]*\[Lambda]^(n - 2*k), {k, 1, Floor[Divide[n,2]]}, GenerateConditions->None]
Failure Failure
Failed [300 / 300]
Result: .3333333331-1.133974598*I
Test Values: {beta = 3/2, lambda = 1/2*3^(1/2)+1/2*I, b[n] = 1/2*3^(1/2)+1/2*I, beta[n] = 1/2*3^(1/2)+1/2*I, n = 2}


Result: -1.032692071-1.500000002*I
Test Values: {beta = 3/2, lambda = 1/2*3^(1/2)+1/2*I, b[n] = 1/2*3^(1/2)+1/2*I, beta[n] = -1/2+1/2*I*3^(1/2), n = 2}

... skip entries to safe data
 Skipped - Because timed out
24.16.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle B_{n,\chi}(x) = \sum_{k=0}^{n}{n\choose k}B_{k,\chi}x^{n-k}}
B_{n,\chi}(x) = \sum_{k=0}^{n}{n\choose k}B_{k,\chi}x^{n-k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
B[n , chi](x) = sum(binomial(n,k)*(B[k , chi](x))^(n - k), k = 0..n)
 
Subscript[B, n , \[Chi]][x] == Sum[Binomial[n,k]*(Subscript[B, k , \[Chi]][x])^(n - k), {k, 0, n}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
24.16.E12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \BernoullipolyB{n}@{x} = B_{n,\chi_{0}}(x-1)}
\BernoullipolyB{n}@{x} = B_{n,\chi_{0}}(x-1)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
bernoulli(n, x) = B[n , chi[0]]*(x - 1)

BernoulliB[n, x] == Subscript[B, n , Subscript[\[Chi], 0]]*(x - 1)
Failure Failure
Failed [280 / 300]
Result: .5669872980-.2500000000*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n,chi[0]] = 1/2*3^(1/2)+1/2*I, chi[0] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: .4836539647-.2500000000*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n,chi[0]] = 1/2*3^(1/2)+1/2*I, chi[0] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [280 / 300]
Result: Complex[0.5669872981077806, -0.24999999999999997]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n, Subscript[χ, 0]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.4836539647744474, -0.24999999999999997]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 0], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n, Subscript[χ, 0]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
24.16.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerpolyE{n}@{x} = -\frac{2^{1-n}}{n+1}B_{n+1,\chi_{4}}(2x-1)}
\EulerpolyE{n}@{x} = -\frac{2^{1-n}}{n+1}B_{n+1,\chi_{4}}(2x-1)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
euler(n, x) = -((2)^(1 - n))/(n + 1)*B[n + 1 , chi[4]]*(2*x - 1)

EulerE[n, x] == -Divide[(2)^(1 - n),n + 1]*Subscript[B, n + 1 , Subscript[\[Chi], 4]]*(2*x - 1)
Failure Failure
Failed [290 / 300]
Result: 1.866025404+.5000000000*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n+1,chi[4]] = 1/2*3^(1/2)+1/2*I, chi[4] = 1/2*3^(1/2)+1/2*I, n = 1}


Result: 1.038675135+.1666666667*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, x = 3/2, B[n+1,chi[4]] = 1/2*3^(1/2)+1/2*I, chi[4] = 1/2*3^(1/2)+1/2*I, n = 2}

... skip entries to safe data
Failed [290 / 300]
Result: Complex[1.8660254037844388, 0.49999999999999994]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, Plus[1, n], Subscript[χ, 4]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.0386751345948129, 0.16666666666666663]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[χ, 4], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, Plus[1, n], Subscript[χ, 4]], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
Retrieved from "https://lct.wmflabs.org/w/index.php?title=24.16&oldid=58445"