Orthogonal Polynomials - 18.34 Bessel Polynomials
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 18.34.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Besselpolyy{n}@{x}{a} = \genhyperF{2}{0}@@{-n,n+a-1}{-}{-\frac{x}{2}}}
\Besselpolyy{n}@{x}{a} = \genhyperF{2}{0}@@{-n,n+a-1}{-}{-\frac{x}{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x] == HypergeometricPFQ[{- n , n + a - 1}, {-}, -Divide[x,2]]
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Missing Macro Error | Failure | - | Error |
| 18.34.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \genhyperF{2}{0}@@{-n,n+a-1}{-}{-\frac{x}{2}} = \Pochhammersym{n+a-1}{n}\left(\frac{x}{2}\right)^{n}\genhyperF{1}{1}@@{-n}{-2n-a+2}{\frac{2}{x}}}
\genhyperF{2}{0}@@{-n,n+a-1}{-}{-\frac{x}{2}} = \Pochhammersym{n+a-1}{n}\left(\frac{x}{2}\right)^{n}\genhyperF{1}{1}@@{-n}{-2n-a+2}{\frac{2}{x}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | hypergeom([- n , n + a - 1], [-], -(x)/(2)) = pochhammer(n + a - 1, n)*((x)/(2))^(n)* hypergeom([- n], [- 2*n - a + 2], (2)/(x))
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HypergeometricPFQ[{- n , n + a - 1}, {-}, -Divide[x,2]] == Pochhammer[n + a - 1, n]*(Divide[x,2])^(n)* HypergeometricPFQ[{- n}, {- 2*n - a + 2}, Divide[2,x]]
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Error | Failure | - | Error |
| 18.34#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y_{n}(x) = \Besselpolyy{n}@{x}{2}}
y_{n}(x) = \Besselpolyy{n}@{x}{2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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Subscript[y, n][x] == Pochhammer[n + 2 - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - 2 + 2, 2/x]
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Missing Macro Error | Failure | - | Failed [89 / 90]
Result: Complex[-1.200961894323342, 0.7499999999999999]
Test Values: {Rule[n, 1], Rule[x, 1.5], Rule[Subscript[y, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-10.950961894323342, 0.7499999999999999]
Test Values: {Rule[n, 2], Rule[x, 1.5], Rule[Subscript[y, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |
| 18.34#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \theta_{n}(x) = x^{n}y_{n}(x^{-1})}
\theta_{n}(x) = x^{n}y_{n}(x^{-1}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | theta[n](x) = (x)^(n)* y[n]((x)^(- 1)) |
Subscript[\[Theta], n][x] == (x)^(n)* Subscript[y, n][(x)^(- 1)] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 18.34#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y_{n}(x;a,b) = \Besselpolyy{n}@{2x/b}{a}}
y_{n}(x;a,b) = \Besselpolyy{n}@{2x/b}{a} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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Subscript[y, n][x ; a , b] == Pochhammer[n + a - 1, n] (2*x/b/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/2*x/b]
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Translation Error | Translation Error | - | - |
| 18.34#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \theta_{n}(x;a,b) = x^{n}y_{n}(x^{-1};a,b)}
\theta_{n}(x;a,b) = x^{n}y_{n}(x^{-1};a,b) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | theta[n](x ; a , b) = (x)^(n)* y[n]((x)^(- 1); a , b) |
Subscript[\[Theta], n][x ; a , b] == (x)^(n)* Subscript[y, n][(x)^(- 1); a , b] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 18.34.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Besselpolyy{n+1}@{x}{a} = (A_{n}x+B_{n})\Besselpolyy{n}@{x}{a}-C_{n}\Besselpolyy{n-1}@{x}{a}}
\Besselpolyy{n+1}@{x}{a} = (A_{n}x+B_{n})\Besselpolyy{n}@{x}{a}-C_{n}\Besselpolyy{n-1}@{x}{a} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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Pochhammer[n + 1 + a - 1, n + 1] (x/2)^n + 1 Hypergeometric1F1[-n + 1, -2 n + 1 - a + 2, 2/x] == (Subscript[A, n]*x + Subscript[B, n])*Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x]-(Divide[- n*(2*n + a),(n + a - 1)*(2*n + a - 2)])*Pochhammer[n - 1 + a - 1, n - 1] (x/2)^n - 1 Hypergeometric1F1[-n - 1, -2 n - 1 - a + 2, 2/x]
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Missing Macro Error | Aborted | - | Failed [300 / 300]
Result: Complex[-1.0464966909469928, 0.15625000000000006]
Test Values: {Rule[a, -1.5], Rule[n, 1], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-13.266992864557183, -0.13671874999999994]
Test Values: {Rule[a, -1.5], Rule[n, 2], Rule[x, 1.5], Rule[Subscript[A, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[B, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
... skip entries to safe data |
| 18.34.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x^{2}\Besselpolyy{n}''@{x}{a}+(ax+2)\Besselpolyy{n}'@{x}{a}-n(n+a-1)\Besselpolyy{n}@{x}{a} = 0}
x^{2}\Besselpolyy{n}''@{x}{a}+(ax+2)\Besselpolyy{n}'@{x}{a}-n(n+a-1)\Besselpolyy{n}@{x}{a} = 0 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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(x)^(2)* D[Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x], {x, 2}]+(a*x + 2)*D[Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x], {x, 1}]- n*(n + a - 1)*Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x] == 0
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Missing Macro Error | Successful | - | Failed [9 / 54]
Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 2], Rule[x, 1.5]}
Result: Indeterminate
Test Values: {Rule[a, -2], Rule[n, 3], Rule[x, 1.5]}
... skip entries to safe data |
| 18.34.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{\alpha\to\infty}\frac{\JacobipolyP{\alpha}{a-\alpha-2}{n}@{1+\alpha x}}{\JacobipolyP{\alpha}{a-\alpha-2}{n}@{1}} = \Besselpolyy{n}@{x}{a}}
\lim_{\alpha\to\infty}\frac{\JacobipolyP{\alpha}{a-\alpha-2}{n}@{1+\alpha x}}{\JacobipolyP{\alpha}{a-\alpha-2}{n}@{1}} = \Besselpolyy{n}@{x}{a} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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Limit[Divide[JacobiP[n, \[Alpha], a - \[Alpha]- 2, 1 + \[Alpha]*x],JacobiP[n, \[Alpha], a - \[Alpha]- 2, 1]], \[Alpha] -> Infinity, GenerateConditions->None] == Pochhammer[n + a - 1, n] (x/2)^n Hypergeometric1F1[-n, -2 n - a + 2, 2/x]
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Missing Macro Error | Aborted | - | Skipped - Because timed out |