Branched continued fraction representations of ratios of Horn's confluent function H6

In this paper, we derive some branched continued fraction representations for the ratios of the Horn's confluent function H6. 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.