  Consider the Bergman kernel $K^B(z)$ of the domain $\ellip = \{z \in \Comp^n
; \sum_{j=1}^n |z_j|^{2m_j}<1 \}$, where $m=(m_1,\ldots,m_n) \in \Natl^n$ and
$m_n \neq 1$. Let $z^0 \in \partial \ellip$ be any weakly pseudoconvex point,
$k \in \Natl$ the degenerate rank of the Levi form at $z^0$. An explicit
formula for $K^B(z)$ modulo analytic functions is given in terms of the polar
coordinates $(t_1, \ldots, t_k, r)$ around $z^0$. This formula provides
detailed information about the singularities of $K^B(z)$, which improves the
result of A. Bonami and N. Lohou\'e \cite{bol}. A similar result is established
also for the Szeg\"o kernel $K^S(z)$ of $\ellip$.
