Bernhard Beckermann, Mirta Castro Smirnova, Valeri Kaliaguine
A recurrence relation connected to the convergence of vector S-fractions
Key words : Convergence of $S$-fractions, Stieltjes moment problem. Simultaneous approximation.
Classifications: AMS(MOS): 39A10, 41A28, 30E05.

Abstract

Recently, some sufficient and necessary conditions have been given on the convergence of the so-called vector Stieltjes continued fraction of dimension $p$ in terms of the coefficients. In the present paper we aim to continue this study for the case of dimension $2$. In particular, we show that here the convergence is determined by the asymptotics of solutions of a particular three-term recurrence relation, which is closely analyzed.

As a consequence, several new results on the convergence problem for two-dimensional Stieltjes continued fractions are obtained. We finally describe the link to a vector moment problem.