On the numerical stability of the branched continued fraction expansion of the ratio H4(a,d+1;c,d;z)/H4(a,d+2;c,d+1;z)

Continued fractions and their generalization, branched continued fractions, are the effective tools used to study special functions. In this aspect, an important problem of continued fractions and branched continued fractions is the study of their numerical stability. The backward recurrence algorithm is one of the main tools for computing approximants of both continued fraction and branched continued fractions. Like most recursive processes, it is prone to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors made in all the previous cycles. This paper considers numerical stability of branched continued fraction expansion of the one ratio of Horn's hypergeometric functions H 4 in the special case, namely, H 4 ( a , d + 1 ; c , d ; z ) / H 4 ( a , d + 2 ; c , d + 1 ; z ) . For this purpose, the backward recurrence algorithm is investigated. It is proven that under certain conditions on the parameters a , c , and d the some open bi-disc is the set of numerical stability for branched continued fraction expansion, and it is found the estimate of relative rounding error, produced by the backward recurrence algorithm in calculating an n th approximant of this expansion. The results of this paper provide a toolkit for analyzing the numerical stability of algorithms that use branched continued fractions of the studied structure. Error estimates can be used to choose computation parameters, control accuracy, and ensure the reliability of results in applied problems that will use the aforementioned branched continued fractions.