Positive definite symmetric functions on finite dimensional spaces. I. Applications of the Radon transform

An n-dimensional random vector X is said (Cambanis, S., Keener, R., and Simons, G. (1983). J. Multivar. Anal., 13 213-233) to have an [alpha]-symmetric distribution, [alpha] > 0, if its characteristic function is of the form [phi]([xi]1[alpha] + ... + [xi]n[alpha]). Using the Radon transform, integral representations are obtained for the density functions of certain absolutely continuous [alpha]-symmetric distributions. Series expansions are obtained for a class of apparently new special functions which are encountered during this study. The Radon transform is also applied to obtain the densities of certain radially symmetric stable distributions on n. A new class of "zonally" symmetric stable laws on n is defined, and series expansions are derived for their characteristic functions and densities.