On convergence of branched continued fraction expansions of Horn's hypergeometric function $H_3$ ratios

The paper deals with the problem of convergence branched continued fractions two branches branching which are used to approximate ratios Horn's hypergeometric function $H_3(a,b;c;{\bf z})$. case real parameters $c\geq a\geq 0,$ b\geq $c\neq and complex variable ${\bf z}=(z_1,z_2)$ is considered. First, it proved fraction for z}\in G_{\bf h}$, where $G_{\bf h}$ two-dimensional disk. Using this result, sufficient conditions uniform above mentioned on every compact subset domain $\displaystyle H=\bigcup_{\varphi\in(-\pi/2,\pi/2)}G_\varphi,$ \[\begin{split} G_{\varphi}=\big\{{\bf z}\in\mathbb{C}^{2}:&\;{\rm Re}(z_1e^{-i\varphi})<\lambda_1 \cos\varphi,\; |{\rm Re}(z_2e^{-i\varphi})|<\lambda_2 \cos\varphi, \\ &\;|z_k|+{\rm Re}(z_ke^{-2i\varphi})<\nu_k\cos^2\varphi,\;k=1,2;\; &\; |z_1z_2|-{\rm Re}(z_1z_2e^{-2\varphi})<\nu_3\cos^{2}\varphi\big\}, \end{split}\] established.