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