The only reference in my texts is Hill. I found this, but I suspect you already know what's in it.I am looking for either a paper or a book to show that Vector Spherical Harmonics form a complete set. I only need it for Euclidean 3 space. As I have been searching I have noticed that they are used in a lot in physics so If anyone has seen them in a mathematical physics book that would be awesome. I can seem to find many math books that talk about Spherical harmonics very much.
So far a I have used these references for Scalar Spherical Harmonics
Analysis of spherical symmetries in Euclidean spaces by Claus Muller
Pure and Applied mathematics vol IX orthogonal functions bu Sansone , G
and I have reference the standard Fourier analysis text by Stien and Wiess.
None of these really talk about Vector Spherical harmonics. However, I
I did find two journal articles in the Library
The Theory of Vector Spherical Harmonics , E.L Hill 1954 and
Vector spherical harmonics and their applications to magnetostatics, Barrera et al. 1984.
This 2nd paper claims that they are complete but does not supply the proof and a rundown of its references has not turned anything up.