New classes of domains with explicit Bergman kernel

We introduce two classes of egg type domains, built on general bounded sym- metric domains, for which we obtain the Bergman kernel in explicit formulas. As an aux- iliary tool, we compute the integral of complex powers of the generic norm on a bounded symmetric domains using the well-known integral of Selberg. This generalizes matrix in- tegrals of Hua and leads to a special polynomial with integer or half-integer coefficients attached to each irreducible bounded symmetric domain.