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Abstract
In this paper, we study expansions for the Dirac operator D, the modified Dirac operator \({D-\lambda,}\) and the polynomial Dirac operator P(D) in super spinor space. These expansions are a meaningful generalization of the classical Almansi expansion in polyharmonic functions theory. As an application of the expansions, the generalized Riquier problem in super spinor space is investigated.
Highlights
In 2013, Coulembier constructed the spinor representation for the orthosymplectic superalgebra osp(m|2n), Sm|2n, which generalizes the so(m)spinors and the symplectic spinors for sp(2n)
We investigate Almansi type expansions in super spinor space
We have studied Almansi expansions for the Dirac operator and the Laplace operator in superspace
Summary
In 2013, Coulembier constructed the spinor representation for the orthosymplectic superalgebra osp(m|2n), Sm|2n (see [5]), which generalizes the so(m)spinors (see [9]) and the symplectic spinors for sp(2n) (see [12]). The Dirac operator is the natural extension of both the classical Dirac operator, for the case n = 0, which acts on the functions defined Rm with values in the orthogonal spinors Sm (see [8]), and the symplectic Dirac operator, for the case m = 0, which acts on sp(2n) on differential forms on R2n with values in the symplectic spinors S0|2n (see [13]) They defined a Laplace operator in super spinor space and studied Fischer decomposition (that is, arbitrary polynomials can be decomposed into a sum of products of the powers of the vector variable with spherical monogenics). We investigate the generalized Riquier problem in super spinor space by the expansion for the operator D − λ
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