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Some integral representation for meta-monogenic function in Clifford algebras depending on parameters. (English) Zbl 1292.15024

The authors begin by generalizing the classical Clifford algebra to one depending on constant parameters, denoted by \(A_{n,2}^*\). They then generalize meta-analytic functions to meta-monogenic (m-m) functions. Let \(\Omega\) be a region of a real (\(n+1\))-space with sufficiently smooth boundary. A continuously differentiable function \(u\) on \(\Omega\) with values in \(A_{n,2}^*\) is called left meta-monogenic on \(\Omega\) if there exist positive integers \(p, m_1, \dots, m_p\) and pairwise distinct real numbers \(\lambda_1, \dots, \lambda_p\) such that \(u\) satisfies the partial differential equation \(\prod_{k=1}^p(D-\lambda_k)^{m_k}u = 0\) on \(\Omega\), where \(D\) is the Dirac operator (and right m-m if the composite operator follows \(u\)). Let the conjugate operator \(\bar{D}\) define anti-meta-monogenic functions. In this paper the authors present integral representation formulas and higher-order Cauchy-Pompeiu-type formulas for the operators \((D-\lambda)kw\) and \((\bar{D}-\lambda)kw\). The last section gives several applications of these results.

MSC:

15A66 Clifford algebras, spinors
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