Geodesic
On the Convergence of Continued T-Fractions
Informatics and Automation, Analytic and geometric issues of complex analysis, Tome 235 (2001), pp. 36-51.

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It is shown that a continued $\mathrm T$-fraction converges on the set $\{|z|$. 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 $\Gamma$. 
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V. I. Buslaev. On the Convergence of Continued T-Fractions. Informatics and Automation, Analytic and geometric issues of complex analysis, Tome 235 (2001), pp. 36-51. http://geodesic.mathdoc.fr/item/TRSPY_2001_235_a1/

[1] Dzhouns U., Tron V., Nepreryvnye drobi, Mir, M., 1985 | MR

[2] Worpitsky J., “Untersuchungen über die Entwickelung der monodromen und monogenen Funktionen durch Kettenbruche”, Friedrichs-Gymnas. und Realsch. Jahresber, Berlin, 1865, 3–39

[3] Van Vlek E., “On the convergence of algebraic continued fractions whose coefficients have limiting values”, Trans. Amer. Math. Soc., 5 (1904), 253–262 | DOI | MR | Zbl

[4] Perron O., Die Lehre von den Kettenbruchen, 2, Teubner, Stuttgart, 1957, 89 | MR

[5] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[6] Buslaev V. I., “O teoreme Van Fleka dlya pravilnykh $\mathrm{C}$-drobei s predelno periodicheskimi koeffitsientami”, Izv. RAN. Ser. mat., 65:4 (2001), 35–48 | MR | Zbl

[7] Polya G., “Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete, III”, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl., 1929, 55–62 | Zbl

[8] Fekete M., “Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten”, Math. Ztschr., 17 (1923), 228–249 | DOI | MR | Zbl

[9] Leja F., “Une generalisation de l'ecart et du diametre transfini d'un ensemble”, Ann. Soc. Polon. Math., 22 (1949), 35–42 | MR

[10] Leja F., “Propriétés des points extrémaux des ensembles plans et leur application à la représentation conforme”, Ann. Polon. Math., 3 (1957), 319–342 | MR

[11] Gonchar A. A., Rakhmanov E. A., “Ravnovesnaya mera i raspredelenie nulei ekstremalnykh polinomov”, Mat. sb., 125 (1984), 117–127 | MR

[12] Mhaskar H. N., Saff E. B., “Weighted analogues of capacity, transfinite diameter, and Chebyshev constant”, Constr. Approx., 8 (1992), 105–124 | DOI | MR | Zbl

[13] Saff E. B., Totik V., Logarithmic potentials with external fields, Grundl. Math. Wissensch., 316, Springer, Berlin, 1997 | MR | Zbl