Painlevé Transcendents - 32.15 Orthogonal Polynomials
| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 32.15.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-\infty}^{\infty}\exp@{-\tfrac{1}{4}\xi^{4}-z\xi^{2}}p_{m}(\xi)p_{n}(\xi)\diff{\xi} = \Kroneckerdelta{m}{n}}
\int_{-\infty}^{\infty}\exp@{-\tfrac{1}{4}\xi^{4}-z\xi^{2}}p_{m}(\xi)p_{n}(\xi)\diff{\xi} = \Kroneckerdelta{m}{n} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | int(exp(-(1)/(4)*(xi)^(4)- z*(xi)^(2))*p[m](xi)* p[n](xi), xi = - infinity..infinity) = KroneckerDelta[m, n]
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Integrate[Exp[-Divide[1,4]*\[Xi]^(4)- z*\[Xi]^(2)]*Subscript[p, m][\[Xi]]* Subscript[p, n][\[Xi]], {\[Xi], - Infinity, Infinity}, GenerateConditions->None] == KroneckerDelta[m, n]
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Failure | Failure | Failed [300 / 300] Result: -.5089051774+.3195154069*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, p[m] = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 1}
Result: .4910948226+.3195154069*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, p[m] = 1/2*3^(1/2)+1/2*I, p[n] = 1/2*3^(1/2)+1/2*I, m = 1, n = 2}
... skip entries to safe data |
Failed [300 / 300]
Result: Complex[-0.5089051767265081, 0.31951540648426185]
Test Values: {Rule[m, 1], Rule[n, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[0.49109482327349185, 0.31951540648426185]
Test Values: {Rule[m, 1], Rule[n, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, m], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |
| 32.15.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a_{n+1}(z)p_{n+1}(\xi) = \xi p_{n}(\xi)-a_{n}(z)p_{n-1}(\xi)}
a_{n+1}(z)p_{n+1}(\xi) = \xi p_{n}(\xi)-a_{n}(z)p_{n-1}(\xi) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | a[n + 1](z)* p[n + 1](xi) = xi*p[n](xi)- a[n](z)* p[n - 1](xi) |
Subscript[a, n + 1][z]* Subscript[p, n + 1][\[Xi]] == \[Xi]*Subscript[p, n][\[Xi]]- Subscript[a, n][z]* Subscript[p, n - 1][\[Xi]] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 32.15.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (u_{n+1}+u_{n}+u_{n-1})u_{n} = n-2zu_{n}}
(u_{n+1}+u_{n}+u_{n-1})u_{n} = n-2zu_{n} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | (u[n + 1]+ u[n]+ u[n - 1])*u[n] = n - 2*z*u[n] |
(Subscript[u, n + 1]+ Subscript[u, n]+ Subscript[u, n - 1])*Subscript[u, n] == n - 2*z*Subscript[u, n] |
Skipped - no semantic math | Skipped - no semantic math | - | - |