The first two coefficients of the Bergman function expansions for Cartan-Hartogs domains

Let $\phi$ be a globally defined real K\"{a}hler potential on a domain $\Omega\subset \mathbb{C}^d$, and $g_{F}$ be a K\"{a}hler metric on the Hartogs domain $M=\{(z,w)\in \Omega\times\mathbb{C}^{d_0}: \|w\|^2<e^{-\phi(z)}\}$ associated with the K\"{a}hler potential $\Phi_{F}(z,w)=\phi(z)+F(\phi(z)+\ln\|w\|^2)$. Firstly, we obtain explicit formulas of the coefficients $\mathbf{a}_j\;(j=1,2)$ of the Bergman function expansion for the Hartogs domain $( M,g_F)$ in a momentum profile $\varphi$. Secondly, using explicit expressions of $\mathbf{a}_j\;(j=1,2)$, we obtain necessary and sufficient conditions for the coefficients $\mathbf{a}_j\;(j=1,2)$ to be constants. Finally, we obtain all the invariant complete K\"{a}hler metrics on Cartan-Hartogs domains such that their the coefficients $\mathbf{a}_j\; (j=1,2)$ of the Bergman function expansions are constants.