About the calculation of hypergeometric function F4 (1,2;2,2; z1, z2 ) by the branched continued fraction of a special kind

Abstract

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.

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

“…Obtained BCF has 2n elements c i(k) z i k , where i(k) ∈ I, in the n-th level. The authors is announced (without proving) the theorem about expansion for hypergeometric function F 4 (1, 2; 2, 2; z 1 , z 2 ) into the branched continued C-fraction of the specific form in the paper [20] . We will prove this theorem and will analyze the structure obtained BCF in this paper.…”