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http://functions.wolfram.com/05.10.21.0003.01
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Integrate[((n - k) (n + k + 1) - (m^2 - l^2)/Sin[\[CurlyTheta]]^2)
Sin[\[CurlyTheta]] SphericalHarmonicY[k, l, \[CurlyTheta], \[CurlyPhi]]
SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]], \[CurlyTheta]] ==
(l - m) Cos[\[CurlyTheta]] SphericalHarmonicY[n, m, \[CurlyTheta],
\[CurlyPhi]] SphericalHarmonicY[k, l, \[CurlyTheta], \[CurlyPhi]] +
(Sin[\[CurlyTheta]] (((Sqrt[Gamma[k - l + 1]] Sqrt[Gamma[k + l + 2]])/
(Sqrt[Gamma[k + l + 1]] Sqrt[Gamma[k - l]])) SphericalHarmonicY[n, m,
\[CurlyTheta], \[CurlyPhi]] SphericalHarmonicY[k, l + 1,
\[CurlyTheta], \[CurlyPhi]] -
((Sqrt[Gamma[n - m + 1]] Sqrt[Gamma[n + m + 2]])/(Sqrt[Gamma[n + m + 1]]
Sqrt[Gamma[n - m]])) SphericalHarmonicY[n, m + 1, \[CurlyTheta],
\[CurlyPhi]] SphericalHarmonicY[k, l, \[CurlyTheta], \[CurlyPhi]]))/
E^(I \[CurlyPhi])
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){kind=small}
