Generalization of the bergman's kernel function

Abstract

A new function of the multiply-connected bounded domains on the plane is introduced that gives generalization of the well-known Bergman's kernel function to the class of polyanalytic functions. The many properties of this function such that reproducing and integral representation are derived. Moreover the important connection between the integral operators with introduced function in the kernel and the singular integral operators on bounded domains is established. In this paper we introduce the generalized kernel function of the bounded domain and give some of its applications. The idea of the Bergman's kernel function is so beautiful and especially its applications are so rich that we do not try to give all those applications, but we give application also to the theory of system of singular integral equations on bounded plane domain that arises in the theory of boundary value problems for the higher-order systems of partial differential equations on the plane.