Bergman Kernel Function On A Class Of Reinhardt Domains And Lu Qikeng Guess

Bergman kernel function is very important in Several Complex Variables, it plays an important role in differential geometry, functional analysis, functional theory and so on. It is known that all bounded domains in (?) have Bergman kernel function, but it is not easy to obtain it. So many mathematicians have been working on it.In this paper we consider a special domain, its definition is as follows:In the second part, using the complete orthonormal system and the holomor-phic automorphism group of the domain, then through some special equalities of Γ function and some computational skills, we obtain the Bergman kernel function on H(1,m,n;K, P). Then we use inflation theory, we obtain the Bergman kernel function on H(N, m, n; K, P).Lu Qi-Keng problem comes on a problem in Lu's paper " The Kahler manifolds with constant curvature ". That problem is called Lu Qi-keng conjecture firstly in 1969 by M. Skwarczynski. If the Bergman kernel function on a domain has no zeros, the domain is called Lu Qi-keng domain. Recently, if we want to research Lu Qi-kcng conjecture, we must firstly obtain the Bergman kernel function on a domain, then use some special skills to judge the presence or absence of zeros of the Bergman kernel function.In the paper, we firstly obtain the Bergman kernel function on H(N, m, n; K, P), then we transform Lu Qi-keng conjecture into a problem about zeros of a polynomia. Using Routh-Hurwitz theorem, we give some results about the presence or absenceof zeros of the Bergman kernel function on H(N, 1,1; K, P), H(N, 1,2; K, P) and H(N,2,1;K,P).