Spherical harmonic analysis for multivariate stable distributions

Series representations consisting of spherical harmonics are obtained for characteristic exponents and probability density functions of multivariate strictly stable distributions under various conditions. A result potentially applicable in a practical setting is that for strictly stable distributions with polynomial spectral spherical densities, the series representations converge absolutely with all their terms being calculable in closed form. Asymptotic expansions consisting of spherical harmonics are also considered for probability density functions.