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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.

##### MSC:
 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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