Fourier Kernels Associated with the Clifford–Helmholtz System

In this paper we present a family of solutions of the Clifford–Helmholtz system, which factors the standard Helmholtz equation. All these solutions can be used as integral kernels of generalized Fourier transforms in hypercomplex analysis. We show that in the Laplace domain they have interesting expressions in terms of terminating hypergeometric functions. This allows us to compute recursion relations between the different kernels. Moreover, we show that it is now possible to compute generating functions of the kernels in different dimensions as well. As a consequence we obtain the generating function of the Clifford–Bessel transform.