hypergeometric series

On convergence of function F4(1,2;2,2;z1,z2) expansion into a branched 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

On the constraction of local gravitational field model

Methods of local gravitational field presentation using nonorthogonal functions are considered. Analysis of SCHA, ASHA and TOSCHA techniques of local field modeling on spherical cap and spherical segment is accomplished according to density distribution of initial data. Approximate formula for finding eigenvalues of differential equation of associated spherical functions   is found and compared with other formulas. This approach involves using of  associated Legendre functions of integer degree and noninteger order. These functions form two sets of functions.