Automorphic Forms in Clifford Analysis

Abstract

In this paper we discuss the possibility of extending the classical theory of automorphic forms to Clifford analysis within the framework of its regularity concepts. To several weights we construct with special functions from Clifford analysis Clifford-valued automorphic forms in a hypercomplex variable that are solutions of iterated homogeneous Dirac equations in $ {\\shadR}^n $ , in particular, generalizations of the classical Eisenstein series and Poincaré series on the upper half-space, on spatial octants and on the unit ball within classes of polymonogenic functions.