Branched continued fraction representations of ratios of Horn`s confluent function $\\mathrm{H}_6$

Authors : Tamara ANTONOVA, Roman DMYTRYSHYN, Serhii SHARYN

Pages : 22-37

Doi:10.33205/cma.1243021

View : 12 | Download : 9

Publication Date : 2023-03-15

Article Type : Research Paper

Abstract :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 $\\Omega$ insert ignore into journalissuearticles values(here, region is a domain insert ignore into journalissuearticles values(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.
Keywords : hypergeometric function, branched continued fraction, convergence