A Chinese Structure On The Bergman Kernel Function

In this paper we consider the Hua Construction of the first type, which is given by Yin Weiping, its definition is as follow:here p_l, q_l > 0, , 1 = 1, ... , r. R_I is Cartan domainof the first type in the sense of Hua, Z denotes the conjugate of Z while Z^t denotes the transpose of Z, det denotes the determinant of a square matrix; N₁, ... , N_t, r are all positive integers.We main get four results: By introducing the concept of semi-Reinhardt domain, using the complete orthonormal system and the holomorphic automorphism group of the Hua Construction of the first type, then through some special equalities of Γ function and some computational skill, we could obtain when 1/(p₁),... ,1/(p_r-1) are positive integers, p_r is any positive real number, the finite sum of the Bergman kernel function on Hua Construction of the first type with explicit formula; by some computating art and the knowlege from the reference, we can also write the Bergman kernel function on Hua Construction of the first type in the form of Appell's hypergeometric functions of several variables while p₁,... ,p_r are all positive integers; furthermore, the infinite series form of the Bergman function on HC_I(N₁,..., N₄, m, n;p₁, ... ,p_r;q₁,... ,q_r) in the general situation can be given; finially, using this result, we could get the explicit formula of the weighted Bergman kernel function on Cartan domain of the first type, here the weighted function is just some negative power of the Bergman function on Cartan domain of the first type.When q₁ = ... q_r = 1, Hua Construction is Hua domain, especially, if m =...