Stirling Numbers and Spin-Euler Polynomials

Abstract

The Fischer decomposition on $\mR^n$ gives the decomposition of arbitrary homogeneous polynomials in $n$ variables $(x_1,\dotsc,x_n)$ in terms of harmonic homogeneous polynomials. In classical Clifford analysis a refinement was obtained, giving a decomposition in terms of monogenic polynomials, i.e., homogeneous null solutions for the Dirac operator (a vector-valued differential operator factorizing the Laplacian $\Delta_n$ on $\mR^n$). In this paper the building blocks for the Fischer decomposition in the Hermitian Clifford setting are determined, yielding a new refinement of harmonic analysis on $\mR^{2n}$ involving complex Dirac operators commuting with the action of the unitary group.