Branched continued fraction representations of ratios of Horn's confluent function $\mathrm{H}_6$

In this paper, we derive some branched continued fraction representations for the ratios of Horn's confluent function $\mathrm{H}_6.$ The method employed is a two-dimensional generalization classical constructing Gaussian fraction. We establish estimates rate convergence expansions in region $\Omega$ (here, domain (open connected set) together with all, part or none its boundary). It also proved that corresponding fractions uniformly converge to holomorphic functions on every compact subset $\Theta,$ and these are analytic continuations double hypergeometric series $\Theta.$ At end, several numerical experiments represented indicate power efficiency as an approximation tool compared series.