branched continued fraction

The paper establishes the conditions of numerical stability of a numerical branched continued fraction using a new method of estimating the relative errors of the computing of approximants using a backward recurrence algorithm.  Based this, the domain of numerical stability of branched continued fractions, which are expansions of Horn's confluent hypergeometric functions $\mathrm{H}_{6} $ with real parameters, is constructed.  In addition, the behavior of the relative errors of computing the approximants of branched continued fraction using the backward recurrence algor

In the paper, the possibility of the Appell hypergeometric function ${F_4}(1,2; 2,2;{z_1},{z_2})$ approximation by a branched continued fraction of a special form is analysed.  The correspondence of the constructed branched continued fraction to the Appell hypergeometric function $F_4$ is proved.  The convergence of the obtained branched continued fraction in some polycircular domain of two-dimensional complex space is established, and numerical experiments are carried out.  The results of the calculations confirmed the efficiency of approximating the Appell hypergeomet