Geodesic
Convergence of the Rogers–Ramanujan continued fraction
Sbornik. Mathematics, Tome 194 (2003) no. 6, pp. 833-856 Cet article a éte moissonné depuis la source Math-Net.Ru

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Set $q=\exp(2\pi i\tau)$, where $\tau$ is an irrational number, and let $R_q$ be the radius of holomorphy of the Rogers–Ramanujan function $$ G_q(z)=1+\sum_{n=1}^\infty z^n\frac{q^{n^2}}{(1-q)\dotsb(1-q^n)}\,. $$ As is known, $R_q\leqslant 1$ and for each $\alpha\in[0,1]$ there exists $q=q(\alpha)$ such that $R_{q(\alpha)}=\alpha$. It is proved here that the function $H_q(z)=G_q(z)/G_q(qz)$ is meromorphic not only in the disc $=\{|z|, but also in the disc $D=\{|z|<1\}$, which is larger for $R_q<1$; and that the Rogers–Ramanujan continued fraction converges to $H_q$ on compact subsets contained in $D\setminus\Omega_q$, where $\Omega_q$ is the union of circles with centres at $z=0$ and passing through the poles of $H_q$. The convergence of the Rogers–Ramanujan continued fraction in the domain $\Bigl\{|z|<\max\bigl(R_q,\frac1{2+|1+q|}\bigr)\Bigr\}\setminus\Omega_q$ was established earlier by Lubinsky. 
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     author = {V. I. Buslaev},
     title = {Convergence of the {Rogers{\textendash}Ramanujan} continued fraction},
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     year = {2003},
     volume = {194},
     number = {6},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2003_194_6_a2/}
}
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V. I. Buslaev. Convergence of the Rogers–Ramanujan continued fraction. Sbornik. Mathematics, Tome 194 (2003) no. 6, pp. 833-856. http://geodesic.mathdoc.fr/item/SM_2003_194_6_a2/

[1] Lubinsky D. S., “Rogers–Ramanujan and the Baker–Gammel–Wills (Padé) conjecture”, Ann. of Math. (2), 157:3 (2003), 847–889 | DOI | MR | Zbl

[2] Baker G. A., Gammel J. L., Wills J. G., “An investigation of the applicability of the Padé approximant method”, J. Math. Anal. Appl., 2 (1961), 405–418 | DOI | MR | Zbl

[3] Stahl H., “Conjectures around Baker–Gammel–Wills conjecture”, Constr. Approx., 13 (1997), 287–292 | DOI | MR | Zbl

[4] Buslaev V. I., “O gipoteze Beikera–Gammelya–Uillsa v teorii approksimatsii Pade”, Matem. sb., 193:6 (2002), 25–38 | MR | Zbl

[5] Hardy G. H., Littlewood J. E., “Notes on the theory of series. XXIV. A curious power-series”, Math. Proc. Cambridge Philos. Soc., 42 (1946), 85–90 | DOI | MR | Zbl

[6] Lubinsky D. S., “Note on polynomial approximation of monomials and diophantine approximation”, J. Approx. Theory, 43 (1985), 29–35 | DOI | MR | Zbl

[7] Petruska G., “On the radius of convergence of $q$-series”, Indag. Math. (N.S.), 3:3 (1992), 353–364 | DOI | MR | Zbl

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

[9] Worpitsky J., “Untersuchungen über die Entwickelung der monodromen und monogenen Funktionen durch Kettenbruche”, Friedrichs-Gymnasium rund Realschule Jahresbericht, Berlin, 1865, 3–39

[10] Polya G., “Über gewisse notwendige Determinantkriterien fur die Forsetzbarkeit einer Potenzreihe”, Math. Ann., 99 (1928), 687–706 | DOI | MR | Zbl

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

[12] Hirschhorn M. D., “Partitions and Ramanujan's continued fraction”, Duke Math. J., 39 (1972), 789–791 | DOI | MR | Zbl

[13] Gasper G., Rahman M., Basic hypergeometric series, Cambridge Univ. Press, Cambridge, 1990 | MR | Zbl

[14] Buslaev V. I., “O teoreme Puankare i ee prilozheniyakh k voprosam skhodimosti tsepnykh drobei”, Matem. sb., 189:12 (1998), 13–28 | MR | Zbl