On the analytic extension of three ratios of Horn's confluent hypergeometric function $\mathrm{H}_7$

Abstract

In this paper, we consider the extension of the analytic functions of two variables by special families of functions — continued fractions. In particular, we establish new symmetric domains of the analytical continuation of three ratios of Horn's confluent hypergeometric function $\mathrm{H}_7$ with certain conditions on real and complex parameters using their continued fraction representations. We use Worpitzky's theorem, the multiple parabola theorem, and a technique that extends the convergence, already known for a small domain, to a larger domain to obtain domains of convergence of continued fractions, and the PC method to prove that they are also domains of analytical continuation.