In this paper we deal with a new class of Clifford algebra valued automorphic forms on arithmetic subgroups of the Ahlfors-Vahlen group. The forms that we consider are in the kernel of the operator D Delta(k/2) for some even k is an element of Z. They will be called k-holomorphic Cliffordian automorphic forms. k-holomorphic Cliffordian functions are well equipped with many function theoretical tools. Furthermore, the real component functions also have the property that they are solutions to the homogeneous and inhomogeneous Weinstein equations. This function class includes the set of k-hypermonogenic functions as a special subset. While we have not been able so far to propose a construction for non-vanishing k-hypermonogenic cusp forms for k not equal 0, we are able to do so within this larger set of functions. After having explained their general relation to hyperbolic harmonic automorphic forms, we turn to the construction of Poincare series. These provide us with non-trivial examples of cusp forms within this function class. Then we establish a decomposition theorem of the spaces of k-holomorphic Cliffordian automorphic forms in terms of a direct orthogonal sum of the spaces of k-hypermonogenic Eisenstein series and of k-holomorphic Cliffordian cusp forms

Abstract. By applying a reflection principle we set up fully explicit representation formulas for the harmonic Green's function for orthogonal sectors of the annulus of the unit ball of R n . From the harmonic Green's function we then can determine the Bergman kernel function of Clifford holomorphic functions by applying an appropriate vector differentiation. As a concrete application we give an explicit analytic representation formula of the solutions to an n-dimensional Dirichlet problem in annular shaped domains that arises in the context of heat conduction.

In this paper we consider inhomogeneous generalized Helmholtz type equations (∆ + λ 2 )u = f in the annulus of an arbitrary ball in R 3 with given boundary conditions, where we assume that λ is an arbitrary complex number. Applying the hypercomplex operator calculus, one can express the solutions in terms of quaternionic integral operators. The quantitative entities to be determined in order to calculate these integral operators in practice are the Cauchy kernel and the Bergman kernel for eigensolutions to the Dirac operator in R 3 . In contrast to the Cauchy kernel which is universal for all domains in R 3 , the Bergman kernel however depends on the domain. In this paper we give an explicit formula for the Bergman kernel of the annulus of a ball in R 3 with arbitrary radii 0 < R1 < R2 < +∞ in terms of explicit special functions. With the knowledge of the Bergman kernel the appearing integral operators can be evaluated fully analytically and thus provide us with explicit formulas for the solutions.

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