Orthogonal Polynomials - 18.25 Wilson Class: Definitions
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 18.25#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle -\delta-1 < \beta}
-\delta-1 < \beta |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | - delta - 1 < beta |
- \[Delta]- 1 < \[Beta] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 18.25#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle N-1 < -\delta-1}
N-1 < -\delta-1 |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | N - 1 < - delta - 1 |
N - 1 < - \[Delta]- 1 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 18.25#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \gamma,\delta > -1,\quad\beta}
\gamma,\delta > -1,\quad\beta |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | gamma , delta > - 1 |
\[Gamma], \[Delta] > - 1 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 18.25#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \gamma,\delta > -1,\quad\beta}
\gamma,\delta > -1,\quad\beta |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | gamma , delta > - 1 |
\[Gamma], \[Delta] > - 1 |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 18.25#Ex5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle N-1 < N+\gamma}
N-1 < N+\gamma |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | N - 1 < N + gamma |
N - 1 < N + \[Gamma] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 18.25#Ex6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle N+\gamma < \beta}
N+\gamma < \beta |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | N + gamma < beta |
N + \[Gamma] < \[Beta] |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 18.25#Ex7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \gamma,\delta < -N,\quad\beta}
\gamma,\delta < -N,\quad\beta |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | gamma , delta < - N |
\[Gamma], \[Delta] < - N |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 18.25#Ex8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \gamma,\delta < -N,\quad\beta}
\gamma,\delta < -N,\quad\beta |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | gamma , delta < - N |
\[Gamma], \[Delta] < - N |
Skipped - no semantic math | Skipped - no semantic math | - | - |
| 18.25.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(y^{2}) = \frac{1}{2y}\left|\frac{\prod_{j}\EulerGamma@{a_{j}+iy}}{\EulerGamma@{2iy}}\right|^{2}}
w(y^{2}) = \frac{1}{2y}\left|\frac{\prod_{j}\EulerGamma@{a_{j}+iy}}{\EulerGamma@{2iy}}\right|^{2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | w((y)^(2)) = (1)/(2*y)*(abs((product(GAMMA(a[j]+ I*y), j = - infinity..infinity))/(GAMMA(2*I*y))))^(2)
|
w[(y)^(2)] == Divide[1,2*y]*(Abs[Divide[Product[Gamma[Subscript[a, j]+ I*y], {j, - Infinity, Infinity}, GenerateConditions->None],Gamma[2*I*y]]])^(2)
|
Failure | Failure | Error | Skip - No test values generated |
| 18.25.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle w(y^{2}) = \frac{1}{2y}\left|\frac{\prod_{j}\EulerGamma@{a_{j}+iy}}{\EulerGamma@{2iy}}\right|^{2}}
w(y^{2}) = \frac{1}{2y}\left|\frac{\prod_{j}\EulerGamma@{a_{j}+iy}}{\EulerGamma@{2iy}}\right|^{2} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | w((y)^(2)) = (1)/(2*y)*(abs((product(GAMMA(a[j]+ I*y), j = - infinity..infinity))/(GAMMA(2*I*y))))^(2)
|
w[(y)^(2)] == Divide[1,2*y]*(Abs[Divide[Product[Gamma[Subscript[a, j]+ I*y], {j, - Infinity, Infinity}, GenerateConditions->None],Gamma[2*I*y]]])^(2)
|
Failure | Failure | Error | Skip - No test values generated |
| 18.25.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \omega_{y} = \frac{\Pochhammersym{\alpha+1}{y}\Pochhammersym{\beta+\delta+1}{y}\Pochhammersym{\gamma+1}{y}\Pochhammersym{\gamma+\delta+2}{y}}{\Pochhammersym{-\alpha+\gamma+\delta+1}{y}\Pochhammersym{-\beta+\gamma+1}{y}\Pochhammersym{\delta+1}{y}y!}}
\omega_{y} = \frac{\Pochhammersym{\alpha+1}{y}\Pochhammersym{\beta+\delta+1}{y}\Pochhammersym{\gamma+1}{y}\Pochhammersym{\gamma+\delta+2}{y}}{\Pochhammersym{-\alpha+\gamma+\delta+1}{y}\Pochhammersym{-\beta+\gamma+1}{y}\Pochhammersym{\delta+1}{y}y!} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | omega[y] = (pochhammer(alpha + 1, y)*pochhammer(beta + delta + 1, y)*pochhammer(gamma + 1, y)*pochhammer(gamma + delta + 2, y))/(pochhammer(- alpha + gamma + delta + 1, y)*pochhammer(- beta + gamma + 1, y)*pochhammer(delta + 1, y)*factorial(y))
|
Subscript[\[Omega], y] == Divide[Pochhammer[\[Alpha]+ 1, y]*Pochhammer[\[Beta]+ \[Delta]+ 1, y]*Pochhammer[\[Gamma]+ 1, y]*Pochhammer[\[Gamma]+ \[Delta]+ 2, y],Pochhammer[- \[Alpha]+ \[Gamma]+ \[Delta]+ 1, y]*Pochhammer[- \[Beta]+ \[Gamma]+ 1, y]*Pochhammer[\[Delta]+ 1, y]*(y)!]
|
Failure | Failure | Failed [300 / 300] Result: .3776605936+.3684973106*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, y = -3/2, omega[y] = 1/2*3^(1/2)+1/2*I}
Result: -.9883648104+.7345227146*I
Test Values: {alpha = 3/2, beta = 3/2, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, y = -3/2, omega[y] = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Skipped - Because timed out |
| 18.25.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \omega_{y} = \frac{(-1)^{y}\Pochhammersym{-N}{y}\Pochhammersym{\gamma+1}{y}\Pochhammersym{\gamma+\delta+1}{2}}{\Pochhammersym{N+\gamma+\delta+2}{y}\Pochhammersym{\delta+1}{y}y!}}
\omega_{y} = \frac{(-1)^{y}\Pochhammersym{-N}{y}\Pochhammersym{\gamma+1}{y}\Pochhammersym{\gamma+\delta+1}{2}}{\Pochhammersym{N+\gamma+\delta+2}{y}\Pochhammersym{\delta+1}{y}y!} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | omega[y] = ((- 1)^(y)* pochhammer(- N, y)*pochhammer(gamma + 1, y)*pochhammer(gamma + delta + 1, 2))/(pochhammer(N + gamma + delta + 2, y)*pochhammer(delta + 1, y)*factorial(y))
|
Subscript[\[Omega], y] == Divide[(- 1)^(y)* Pochhammer[- N, y]*Pochhammer[\[Gamma]+ 1, y]*Pochhammer[\[Gamma]+ \[Delta]+ 1, 2],Pochhammer[N + \[Gamma]+ \[Delta]+ 2, y]*Pochhammer[\[Delta]+ 1, y]*(y)!]
|
Failure | Failure | Failed [300 / 300] Result: -3.383353139+40.73029447*I
Test Values: {N = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, y = -3/2, omega[y] = 1/2*3^(1/2)+1/2*I}
Result: -4.749378543+41.09631987*I
Test Values: {N = 1/2*3^(1/2)+1/2*I, delta = 1/2*3^(1/2)+1/2*I, gamma = 1/2*3^(1/2)+1/2*I, omega = 1/2*3^(1/2)+1/2*I, y = -3/2, omega[y] = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Failed [300 / 300]
Result: Complex[4.68201860981384, 5.925892618408873]
Test Values: {Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, y], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[3.3159932060294013, 6.291918022193311]
Test Values: {Rule[N, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5], Rule[γ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[δ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[ω, y], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |