Zonal decompositions for spherical monogenics

Clifford analysis is to be considered as a natural generalization of the theory of complex holomorphic functions in the plane to higher dimension, in which the Dirac operator generalizes the Cauchy-Riernann operator partial derivative(z) = partial derivative(x), + i partial derivative(y.) Null-solutions for this operator are so-called monogenic functions. and the Study of these lies at the very heart of Clifford analysis (see [1-3]). Because the Dirac operator is a vector-valued differential operator factorizing the Laplace operator Delta(m) on R-m, Clifford analysis can be seen as a refinement of harmonic analysis. In this paper we study the spherical monogenics, which are a refinement within the context of Clifford analysis of the classical spherical harmonics in R-m, i.e. homogeneous polynomial null-solutions for Delta(m.) Using a recursive argument, we will construct spherical monogenics of a particular type. For that purpose, we first choose an orthonormal basis {e1,..., e(m)} for R-m, and then construct spherical monogenics as products of zonal building blocks, depending on the inner product with these vectors only.