On the first two coefficients of the Bergman function expansion for radial metrics

Let gF be Kähler metrics on rotation invariant domains Ω=Bd, Cd,Bd∗,Cd∗ associated with the Kähler potentials ΦF(z,z¯)=F(ln(‖z‖2)). The purpose of this paper is twofold. Firstly, we obtain explicit formulas of the coefficients aj(j=1,2) of the Bergman function expansion for the domains (Ω,gF). Secondly, we obtain explicit expressions of F when both a1 and a2 are constants on Ω.