On the Convergence of Continued T-Fractions

It is shown that a continued $\mathrm T$-fraction converges on the set $\{|z|<R_1\} \cup \{|z|>R_2\}$. Formulas (exact in a certain sense) for evaluating the radii $R_1$ and $R_2$ of these disks are given. For a $\mathrm T$-fraction with limit-periodic coefficients, a cut $\Gamma$ on the complex plane is explicitly specified such that this $\mathrm T$-fraction converges outside this cut. It is shown that the meromorphic function represented by this $\mathrm T$-fraction cannot be meromorphically continued (as a single-valued function) across any arc lying on $\Gamma$.