On a Theorem of T. Gneiting on α-Symmetric Multivariate Characteristic Functions

T. Gneiting (1998, J. Multivariate Analysis64, 131–147) proved a relation between the primitives of the classes Φd(2) and Φd(1) of 2- and 1-symmetric characteristic functions on Rd, respectively. We will give a straightforward proof of his relation, answering a question of his. To do this we use the calculus of generalized hypergeometric functions.