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 H4 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⊂C2, 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 H4 is considered, and the expression of solutions of some systems of partial differential equations is indicated.
The paper deals with the problem of approximation of functions of several variables by branched continued fractions. We study the correspondence between formal multiple power series and the so-called "multidimensional $S$-fraction with independent variables". As a result, the necessary and sufficient conditions for the expansion of the formal multiple power series into the corresponding multidimensional $S$-fraction with independent variables have been established. Several numerical experiments show the efficiency, power and feasibility of using the branched continued fractions in order to numerically approximate certain functions of several variables from their formal multiple power series.
The paper investigates the convergence problem of a special class of branched continued fractions, i.e. the multidimensional S-fractions with independent variables, consisting of \[\sum_{i_1=1}^N\frac{c_{i(1)}z_{i_1}}{1}{\atop+}\sum_{i_2=1}^{i_1}\frac{c_{i(2)}z_{i_2}}{1}{\atop+} \sum_{i_3=1}^{i_2}\frac{c_{i(3)}z_{i_3}}{1}{\atop+}\cdots,\] which are multidimensional generalizations of S-fractions (Stieltjes fractions). These branched continued fractions are used, in particular, for approximation of the analytic functions of several variables given by multiple power series. For multidimensional S-fractions with independent variables we have established a convergence criterion in the domain \[H=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|\arg(z_k+1)|<\pi,\; 1\le k\le N\right\}\] as well as the estimates of the rate of convergence in the open polydisc \[Q=\left\{{\bf{z}}=(z_1,z_2,\ldots,z_N)\in\mathbb{C}^N:\;|z_k|<1,\;1\le k\le N\right\}\] and in a closure of the domain $Q.$
In this paper, we derive some branched continued fraction representations for the ratios of the Horn's confluent function $\mathrm{H}_6.$ The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. We establish the estimates of the rate of convergence for the branched continued fraction expansions in some region $\Omega$ (here, region is a domain (open connected set) together with all, part or none of its boundary). It is also proved that the corresponding branched continued fractions uniformly converge to holomorphic functions on every compact subset of some domain $\Theta,$ and that these functions are analytic continuations of the ratios of double confluent hypergeometric series in $\Theta.$ At the end, several numerical experiments are represented to indicate the power and efficiency of branched continued fractions as an approximation tool compared to double confluent hypergeometric series.
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