Holomorphic Cliffordian product (Q1772532)

The class of holomorphic Cliffordian functions was introduced in [\textit{G. Laville} and \textit{I. Ramadanoff}, Holomorphic Cliffordian functions 8, No. 2, 323--340 (1998; Zbl 0940.30028)]. Because of the non-commutativity of multiplication in a Clifford algebra the point-wise product of such functions is not holomorphic Cliffordian already, and the author is concerned with constructing of an adequate product. The basic idea here is to use the fact that the anticommutator of two paravectors is a paravector. Thus in Section 2 the algebraic structure on the paravector space of the Clifford algebra with negative signature is studied which includes: some properties of symmetric products of paravectors; the symmetric algebra of the underlying real \(n\)-dimensional space; interesting examples; symmetrization by integral means. This tool is used in Section 3, Analysis with the holomorphic Cliffordian product. First of all, the properties of the symmetric product of paravectors extend onto the normally convergent series of products of paravectors; then an integral representation is given for holomorphic Cliffordian products together with an analog of the Lagrange interpolation formula; the concluding subsections treat derivatives and Cauchy-Riemann type equations; an analog of the Taylor formula and a kind of the Cauchy integral theorem. All this in the context of paravector-valued holomorphic Cliffordian functions.