Sum[I^(n - m - 1) Sqrt[((2 n + 1) (n + m)!)/(n - m)!] BesselJ[n + 1/2, w] SphericalHarmonicY[n, m, \[CurlyTheta], \[CurlyPhi]], {n, m + 1, Infinity, 2}] == (Sqrt[w]/(Sqrt[2] Pi)) ((-w) Sin[\[CurlyTheta]] E^(I \[CurlyPhi]))^m Sin[w Cos[\[CurlyTheta]]] /; m >= 0 && Element[\[CurlyTheta], Reals] && Element[\[CurlyPhi], Reals]

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2001-10-29