continued fraction

In the paper, the possibility of the Appell hypergeometric function ${F_4}(1,2; 2,2;{z_1},{z_2})$ approximation by a branched continued fraction of a special form is analysed.  The correspondence of the constructed branched continued fraction to the Appell hypergeometric function $F_4$ is proved.  The convergence of the obtained branched continued fraction in some polycircular domain of two-dimensional complex space is established, and numerical experiments are carried out.  The results of the calculations confirmed the efficiency of approximating the Appell hypergeomet

An algorithm for constructing recurrence relations of geometric Gaussian functions, in which the displacement of parameters is equal to $0$, $1$ or $-1$, is described. On the basis of such recurrence relations, the expansion for the ratio of Gaussian functions into continued fractions is developed. The obtained continued fractions are the development of the corresponding hypergeometric Gaussian functions in the case when the parameters of the function are integers.