Differentiability of bizonal positive definite kernels on complex spheres

We prove that any continuous function with domain (z is an element of C: vertical bar z vertical bar <= 1) that generates a bizonal positive definite kernel on the unit sphere in C-q, q >= 3, is continuously differentiable in {z is an element of C: vertical bar z vertical bar < 1} up to order q - 2, with respect to both z and <(z)over bar>. In particular, the partial derivatives of the function with respect to x = Re z and y = Im z exist and are continuous in {z is an element of C: vertical bar z vertical bar < 1} up to the same order. (C) 2013 Elsevier Inc. All rights reserved.