Asymptotic expansion of polyanalytic Bergman kernels

We consider the q-analytic functions on a given planar domain Ω, square integrable with respect to a weight. This gives us a q-analytic Bergman kernel, which we use to extend the Bergman metric to this context. We recall that f is q  -analytic if ∂¯qf=0 for the given positive integer q. Polyanalytic Bergman spaces and kernels appear naturally in time-frequency analysis of Gabor systems of Hermite functions as well as in the mathematical physics of the analysis of Landau levels.We obtain asymptotic formulae in the bulk for the q  -analytic Bergman kernel in the setting of the power weights e−2mQe−2mQ, as the positive real parameter m   tends to infinity. This is only known previously for q=1q=1, by the work of Tian, Yau, Zelditch, and Catlin. Our analysis, however, is inspired by the more recent approach of Berman, Berndtsson, and Sjöstrand, which is based on ideas from microlocal analysis. We remark here that since a q-analytic function may be identified with a vector-valued holomorphic function, the Bergman space of q  -analytic functions may be understood as a vector-valued holomorphic Bergman space supplied with a certain singular local metric on the vectors. Finally, we apply the obtained asymptotics for q=2q=2 to the bianalytic Bergman metrics, and after suitable blow-up, the result is independent of Q for a wide class of potentials Q. We interpret this as an instance of geometric universality.