On the approximation of Lauricella–Saran’s hypergeometric functions FM and their ratios by branched continued fractions

The paper delves the problem of approximating Lauricella–Saran’s hypergeometric functions by a special family of functions – branched continued fractions. Under certain conditions of the parameters of the Lauricella–Saran’s hypergeometric functions FM , new domains of the analytic continuation of these functions and their ratios are established, using their expansions into branched continued fractions, the elements of which are polynomials of three complex variables. At the end, several numerical experiments are presented that illustrate the efficient approximation of special function by branched continued fraction.