On the branched continued fraction expansions of the complete group of ratios of the generalized hypergeometric function $_4F_3$

The paper considers the classical problem of the rational approximation of analytic functions of complex variable, in particulary, to issues that arise when constructing branched continued fraction expansions for generalized hypergeometric functions. Using combinations of three- and four-term recurrence relations of the generalized hypergeometric function $_4F_3$, we constructed the formal branched continued fraction expansions of sixteen ratios of this function. These sixteen ratios are the complete group of ratios of the generalized hypergeometric function $_4F_3$. This means that each of these ratios has a formal branched continued fraction expansion that uses all of these ratios.