  The reproducing kernel function of a weighted Bergman space over domains in
${\mathbb C}^d$ is known explicitly in only a small number of instances. Here,
we introduce a process of orthogonal norm expansion along a subvariety of
codimension 1, which also leads to a series expansion of the reproducing kernel
in terms of reproducing kernels defined on the subvariety. The problem of
finding the reproducing kernel is thus reduced to the same kind of problem when
one of the two entries is on the subvariety. A complete expansion of the
reproducing kernel may be achieved in this manner. We carry this out in
dimension $d=2$ for certain classes of weighted Bergman spaces over the bidisk
(with the diagonal $z_1=z_2$ as subvariety) and the ball (with $z_2=0$ as
subvariety), as well as for a weighted Bargmann-Fock space over ${\mathbb C}^2$
(with the diagonal $z_1=z_2$ as subvariety).
