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In this paper we present applications of a special class of homogeneous monogenic polynomials constructed, in the framework of hypercomplex function theory, in order to be an Appell set of polynomials. In particular, we derive important properties of an associated exponential function from R 3 to R 3 and propose a generalization to R n+1 .

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In Clifford Analysis, several different methods have been developed for constructing monogenic functions as series with respect to properly chosen homogeneous monogenic polynomials. Almost all these methods rely on sets of orthogonal polynomials with their origin in classical (real) Harmonic Analysis in order to obtain the desired basis of homogeneous polynomials. We use a direct and elementary approach to this problem and construct a set of homogeneous polynomials involving only products of a hypercomplex variable and its hypercomplex conjugate. The obtained set is an Appell set of monogenic polynomials with respect to the hypercomplex derivative. Its intrinsic properties and some applications are presented.

Hypercomplex function theory generalizes the theory of holomorphic functions of one complex variable by using Clifford Algebras and provides the fundamentals of Clifford Analysis as a refinement of Harmonic Analysis in higher dimensions. We define the Laguerre derivative operator in hypercomplex context and by using operational techniques we construct generalized hypercomplex monogenic Laguerre polynomials. Moreover, Laguerre-type exponentials of order m are defined.