Weighted Bergman kernels and $\star$-products

We calculate the weighted Bergman kernel on a complex domain with a weight of the form $\rho =e^{-\alpha \phi }\mu g$, where $\alpha$ is a positive real number, $\phi$ is a Kähler potential, g is the determinant of the corresponding Kähler metric and $\mu$ is a real-valued positive function. Several $\star$-products related to the Bergman kernel are determined. Explicit formulas are provided up to first order.