Asymptotic expansion of POLYANALYTIC BERGMAN KERNELS

It is well-known from the work of Tian, Yau, Zelditch, Catlin, that the Bergman kernel with respect to the weight e −mQ has an asymptotic expansion as m → +∞. In the setting of one complex variable, we extend these results to spaces of q-analytic functions, where a function f is q-analytic if∂ q f = 0 for the given positive q. As a q-analytic function may be identified with a vector-valued holomorphic function, the Bergman space of q-analytic functions can be understood as a vector-valued holomorphic Bergman space supplied with a certain singular local metric on the vectors