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 { ( z 1 , z 2 , z 3 ) ∈ C 3 : ∣ z 1 ∣ 4 + ∣ z 2 ∣ 4 + ∣ z 3 ∣ 4 < 1 } and { ( z 1 , z 2 , z 3 ) ∈ C 3 : ∣ z 1 ∣ 4 + ∣ z 2 ∣ 4 + ∣ z 3 ∣ 2 < 1 } . We also investigate the boundary behavior of the Bergman kernel functions near some weakly pseudoconvex boundary points.