A partial differential equations is introduced on the basis of the definitions of k-hypermonogenic function with vector value and the k-hyperbolically harmonic function,then the porperties of k-hypermonogenic function with vector value and their relations are discussed,at last a sufficient and necessary condition for the solvability of partial differential equations is obtained.

Moreover, it is also proved that solutions of the Dirichlet boundary value problems for above two classes of functions and the k-harmonic function uniquely exist.

On the basis of the defination of K-integral,this essay deducts out the relation between K-analytic function and K-integral and the relation between K-analytic function and K-harmonic functions.

In the first part of this paper,resorting to the idea of quasi-permutation which created by Sha Hang,we give the equivalent conditions of complex monogenic and complex hypermonogenic functions,define the complex hyperharmonic function and discuss its equivalent conditions.