On approximation of some Lauricella-Saran's hypergeometric functions $F_M$ and their ratios by branched continued fractions

The paper considers the problem of approximating Lauricella-Saran's hypergeometric functions $F_M(a_1,a_2,b_1,b_2;a_1,c_2;z_1,z_2,z_3)$ by rational functions, which are approximants of branched continued fraction expansions - a special family functions. Under the conditions of positive definite values of the elements of the expansions, the domain of analytic continuation of these functions and their ratios is established. Here, the domain is an open connected set. It is also proven that under the above conditions, every branched continued fraction expansion converges to the function that is holomorphic in a given domain of analytic continuation at least as fast as a geometric series with a ratio less then unity.