Differentiability of bizonal positive definite kernels on complex spheres

We prove that any continuous function with domain { z ∈ C : | z | ⩽ 1 } that generates a bizonal positive definite kernel on the unit sphere in C q , q ⩾ 3 , is continuously differentiable in { z ∈ C : | z | < 1 } up to order q − 2 , with respect to both z and z ¯ . In particular, the partial derivatives of the function with respect to x = Re z and y = Im z exist and are continuous in { z ∈ C : | z | < 1 } up to the same order.