Explicit formulas of the Bergman kernel for 3-dimensional complex ellipsoids

In this paper we obtain the closed forms of some hypergeometric functions with half-integer values of parameters. As an application, we obtain the explicit forms of the Bergman kernel functions for complex ellipsoids {(z1,z2,z3)∈C3:∣z1∣4+∣z2∣4+∣z3∣4<1}{(z1,z2,z3)∈C3:∣z1∣4+∣z2∣4+∣z3∣4<1} and {(z1,z2,z3)∈C3:∣z1∣4+∣z2∣4+∣z3∣2<1}{(z1,z2,z3)∈C3:∣z1∣4+∣z2∣4+∣z3∣2<1}. We also investigate the boundary behavior of the Bergman kernel functions near some weakly pseudoconvex boundary points.