| Let Ω be a bounded domain in Cn, and H2(Ω) is the space of square-integrable holomorphic functions on Ω, in other words, H2(Ω) = Hol(Ω) ∩L2(Ω). We introduce an inner product on Ω: , then H2(Ω) is a Hilbert space. H2(Ω) is a separable nonempty inner product space, so it has countable complete orthonormal basis. Let {φk(z)}, k = 0,1, … be the complete orthonormal basis of H2(Ω), and the Bergman kernel function can be represented byWe know that the Bergman kernel function of Ω is unique, it's independent of the choice of the complete orthonormal basis. We can also define the Bergman kernel function using its reproducing property.From the definition we can see that if the complete orthonormal basis of a domain is given, we can get its infinite series sum of the Bergman kernel function. For some special bounded domain, we can get its Bergman kernel function with explicit formula by some summing skill. So the complete orthonormal basis is very important for us to get the Bergman kernel function with explicit formula. To the bounded domain in Cn, we have known the complete orthonormal basis of the circular domain, the Reinhardt domain and the semi-Reinhardt domain. In the first section of this paper, we will give the definition and the complete orthonormal basis of the grouping-circular domain.The main results of this paper: firstly we introduce the definition of grouping-circular domain and give its complete orthonormal system; secondly, as a use of grouping-circular domian, we obtain the explicit formula of the Bergman kernelfunction on HD, the definition of HD is: HD(1,m1n1,m2n2,p,q1,q2):= {ξ∈C,Z ∈RI(m1,n1), W ∈RI{m2,n2) : |ξ|2p<φ(Z, W)},here , p> 0, q1 > 0, q2 > 0, Z|- denotes the conjugate of Z, Zt denotes the transpose of Z, det denotes the determinant of a square matrix; finally, we genelized our conclusion to domain HD, it is to say, if ξ = (ξ1,…,ξr) ∈ Cr, when 1/p1, … , 1/pr-1 are positive integers, pr is any positive real number, we can get the the finite sum of the Bergman kernel function on HD:,pl > 0, l = 1,2, … , r, q1> 0, q2 > 0).If we write the defined function of the Hua domain(including Egg domain) into the form of Hartogs domain, it will be Hartogs domain, and its base space is irreducible. But the base space of HD is reducible, it's the direct product of two Cartan domains of the first type in the sense of Hua, so we give a method to the computation of the Bergman kernel function on a new kind of Hartogs domain. |
