The weighted reproducing kernels of the Reinhardt domain

In this paper, we develop the theory of weighted Bergman spaces and obtain a general representation formula of the kernel function for the spaces on the Reinhardt domain. As a complement to the general study, we calculate the concrete forms of the Bergman kernels for some special weights on the Reinhardt domain $\mathbb{C}^n,$ $ D_{n,m}:= \{(z, w)\in \mathbb{C}^n \times \mathbb{C}^m : \|w\|^2 < e^{-\mu_1\|z\|^{\mu_2}} \}$ and $V_{\eta}=\left\{\left(z, z', w\right) \in \mathbb{C}^{n} \times \mathbb{C}^{m} \times \mathbb{C} : \sum_{j=1}^{n} e^{\eta_{j}|w|^{2}}\left|z_{j}\right|^{2}+\|z'\|^{2}