Surface harmonic expansions of products of Cartesian coordinates

We use Hobson's definition of the associated Legendre functions [1]. Any product of non-negative powers of (x/r), (y/r), and (z/r) can be expanded as a linear combination of harmonics Pn(cos 0), PFm(cos 0) cos mp and Pnm(cos 0) sin m4 in which n and m take appropriate values. This expansion is given formally by Equation (2), which defines the coefficients a',,,. We consider only non-negative integer values of u, v, w.