On the double series expansion of holomorphic functions

A holomorphic function f in a neighborhood of 0 in Cⁿ⁺¹ can be expanded into the double series: f(z) = ∑_k=0^∞ ∑_l=0^[k/2] (z²)^l f_k,k-2l(z), where f_k,k-2l is a homogeneous harmonic polynomial of degree k - 2l and z² = z₁² + ⋯ + z_n+1². We characterized holomorphic functions on the complex Euclidean ball, on the Lie ball or on the dual Lie ball by the growth behavior of homogeneous harmonic polynomials in their double series expansion. In this paper, we consider holomorphic functions and analytic functionals on an N_p-ball which lies between the Lie ball and the dual Lie ball, and characterize them by the growth behavior of homogeneous harmonic polynomials. Our results lead a new proof of a known theorem on the Fourier-Borel transformation.