A higher order integral representation formula in isotonic Clifford analysis

Abstract

Let ∆ be the Laplace operator in R 2n . The aim of this paper is to present an integral representation formula for the solutions of the generalized isotonic system ∂ 1 ∆ k f + i∆ f ∂ 2 = 0, k ∈ N0, where ∂ 1 , ∂ 2 are Dirac type operators and wher f stands for the main invo-lution in the complex Clifford algebra Cn. Two special cases of this represen-tation are also discussed, yielding generalizations of the Bochner-Martinelli formula for the holomorphic functions and for the biregular functions. Mathematics Subject Classification (2000). 30G35. 1. Polymonogenic functions Let C m be the complex Clifford algebra constructed over the orthonormal basis (e 1 , . . . , e m) of the Euclidean space R m (see [14]). The multiplication in C m is determined by the relations e j e k + e k e j = −2δ jk , j, k = 1, . . . , m, where δ jk is the Kronecker delta. A general element of C m is of the form a =