The multiplication of the Clifford-analytic Eisenstein series

In this paper, we deduce general multiplication formulas for hypercomplex monogenic and polymonogenic generalizations of Eisenstein series that are related to translation groups. In particular, a criterion for paravector multiplication of arbitrary finite-dimensional lattices in terms of being integral is developed. Under these number theoretical conditions it is then possible to transfer the concept of the complex multiplication of the -function to the framework of its hypercomplex higher dimensional analogues within the Clifford analysis setting. We derive explicit formulas for the hypercomplex division values of the hypercomplex monogenic and polymonogenic Eisenstein series for lattices with hypercomplex multiplication. We also provide applications to the function theory. In particular, it will be shown that a non-constant polymonogenic function that satisfies one of the specific integer multiplication formulas of the Clifford-analytic Eisenstein series must have singularities. Furthermore, it coincides with one of those polymonogenic Eisenstein series whenever all the singularities are distributed in the form of a lattice and have all the same order and principal parts.