Representations for Appell's series $F_2(x,y)$ to the vicinity of the singular point $(1,1)$ and near the boundary of its domain of convergence

Exact analytical representations for Appell's series $F_2(x,y)$ to the vicinity of the singular point $(1,1)$ and the boundary of its domain of convergence are given. It is shown, that Appell's functions $F_2(1,1)$ and $F_3(1,1)$ have the property of mirror-like symmetry with respect to the center $j_0=-1/2$ under the change $j\mapsto-j-1$, $j\in\mathbb{Z}$, and they correlate between each other.