Stirling numbers and spin-Euler polynomials. (English) Zbl 1201.30064

Summary: The Fischer decomposition on \(\mathbb R^n\) gives the decomposition of arbitrary homogeneous polynomials in \(n\) variables \(x_1,\ldots,x_n\) 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 \(\mathbb R^n\)). 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 \(\mathbb R^{2n}\) involving complex Dirac operators commuting with the action of the unitary group.


30G35 Functions of hypercomplex variables and generalized variables
32W50 Other partial differential equations of complex analysis in several variables
15A66 Clifford algebras, spinors
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