On Some Branched Continued Fraction Expansions for Horn's Hypergeometric Function H4(a,b;c,d;z1,z2) Ratios

The paper deals with the problem of representation of Horn's hypergeometric functions by branched continued fractions. The formal branched continued fraction expansions for three different Horn's hypergeometric function H-4 ratios are constructed. The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. It is proven that the branched continued fraction, which is an expansion of one of the ratios, uniformly converges to a holomorphic function of two variables on every compact subset of some domain H, H subset of C-2, and that this function is an analytic continuation of this ratio in the domain H. The application to the approximation of functions of two variables associated with Horn's double hypergeometric series H-4 is considered, and the expression of solutions of some systems of partial differential equations is indicated.

Automated summary

This paper deals with the representation problem of Horn's hypergeometric functions using branched continued fractions. The formal branched continued fraction expansions for three different Horn's hypergeometric function H-4 ratios are constructed. A two-dimensional generalization of the classical method of constructing Gaussian continued fractions is employed. It is proven that the branched continued fraction converges uniformly to a holomorphic function of two variables on every compact subset of some domain H, H subset of C-2, and that this function is an analytic continuation of this ratio in the domain H. The application to the approximation of functions of two variables associated with Horn's double hypergeometric series H-4 is considered, and the expression of solutions of some systems of partial differential equations is indicated.