  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.
