Complex symmetric difference of the weighted composition operators on weighted Bergman space of the half-plane

The main goal of this paper was to completely characterize complex symmetric difference of the weighted composition operators induced by three type symbols on weighted Bergman space of the right half-plane with the conjugations $\mathcal{J}f(z)=\overline{f(\overline{z})}$, ${\mathcal{J}}_{s}f(z)=\overline{f(\overline{z}+is)}$, and ${\mathcal{J}}_{\ast }f(z)=\frac{1}{z^{\alpha +2}}\overline{f(\frac{1}{\overline{z}})}$. The special phenomenon that we focus on is that the difference is complex symmetric on weighted Bergman spaces of the half-plane with the related conjugation if and only if each weighted composition operator is complex symmetric.