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

Eelbode, D.

Exp. Math. 16, No. 1, 55-66 (2007).

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

Keywords:

Fischer decomposition; Hermitian Clifford analysis, Stirling numbers