  We consider mainly the Hilbert space of bianalytic functions on a given
domain in the plane, square integrable with respect to a weight. We show how to
obtain the asymptotic expansion of the corresponding bianalytic Bergman kernel
for power weights, under the standard condition on those weights. This is known
only in the analytic setting, from the work of e.g. Tian, Yau, Zelditch,
Catlin, et al. We remark that a bianalytic function may be identified with a
vector-valued analytic function, supplied with a locally singular metric on the
vectors. We also apply our findings to two bianalytic Bergman metrics
introduced here.
