On Analytical Continuation of the Horn’s Hypergeometric Functions H3 and Their Ratios

This paper considers the Horn’s hypergeometric function H3, which is closely related to other hypergeometric functions and has various mathematical or physical applications. The problem of analytical extension of this function is solved using a special family of functions—branched continued fractions. A new domain of analytical extension of the Horn’s hypergeometric functions H3 and their ratios under certain conditions to real parameters are established. This paper also contains an example of the presentation and continuation of some special function and an analysis of numerical results.