In the paper, the possibility of the Appell hypergeometric function F4(1,2;2,2;z1,z2) 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 F4 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 hypergeometric function F4(1,2;2,2;z1,z2) by a branched continued fraction of special form and illustrated the hypothesis of the existence of a wider domain of convergence of the obtained expansion.

Дослiджено вiдповiднiсть, збiжнiсть i стiйкiсть до збурень нескiнченних залишкiв гiллясто-го ланцюгового дробу Ньорлунда в полiкруговiй областi {(z 1 , z 2 ) ∈ C 2 : |z j | ≤ r, j = 1, 2}, 0 < r < 1/8, у випадку довiльних параметрiв гiпергеометричної функцiї Аппеля.Ключовi слова i фрази: гiпергеометрична функцiя Аппеля, гiллястий ланцюговий дрiб.

In the paper, using some recurrent relations, the expansion of the hypergeometric Appel function F4 (1,2;2,2; z1, z2 ) into a branched continued fraction of special form is constructed. Explicit formulas for the coefficients of constructed development are obtained. The structure of the obtained branched continued fraction is investigated. The values of the suitable fractions and the corresponding partial sums of the hypergeometric series at different points of the two-dimensional complex space are calculated. A comparative analysis of the obtained values is carried out, the results of which confirm the efficiency of using branched continued fractions to calculate the values of the hypergeometric function F4 (1,2;2,2; z1, z2 ) in space C2.