The paper investigates the problem of constructing branched continued fraction expansions of hypergeometric functions \(F_M(a_1,a_2,b_1,b_2;a_1,c_2;\mathbf{z})\) and their ratios. Recurrence relations of the hypergeometric function \(F_M\) are established, which provide the construction of formal branched continued fractions with simple structures, the elements of which are polynomials in the variables \(z_1, z_2, z_3.\) To construct the expansions, a method of based on the so-called complete group of ratios of hypergeometric functions was used, which is a generalization of the classical Gauss method.