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 Mat. Sb., 2003, Volume 194, Number 6, Pages 43–66 (Mi msb741)

Convergence of the Rogers–Ramanujan continued fraction

V. I. Buslaev

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: 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 $=\{|z|<R_q\}$, 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 $\{|z|<\max(R_q,\frac1{2+|1+q|})\}\setminus\Omega_q$ was established earlier by Lubinsky.

DOI: https://doi.org/10.4213/sm741

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English version:
Sbornik: Mathematics, 2003, 194:6, 833–856

Bibliographic databases:

UDC: 517.524
MSC: 30B70, 41A21

Citation: V. I. Buslaev, “Convergence of the Rogers–Ramanujan continued fraction”, Mat. Sb., 194:6 (2003), 43–66; Sb. Math., 194:6 (2003), 833–856

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb741
• https://doi.org/10.4213/sm741
• http://mi.mathnet.ru/eng/msb/v194/i6/p43

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Lubinsky D.S., “On Baker'S Patchwork Conjecture For Diagonal Pade Approximants”, Constr. Approx.
2. Lukashov A.L., Peherstorfer F., “Zeros of polynomials orthogonal on two arcs of the unit circle”, J. Approx. Theory, 132:1 (2005), 42–71
3. Buslaev V.I., “On Hankel Determinants of Functions Given by their Expansions in P-Fractions”, Ukrainian Mathematical Journal, 62:3 (2010), 358–372
4. A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Russian Math. Surveys, 66:6 (2011), 1049–1131
5. A. Kuznetsov, “On the density of the supremum of a stable process”, Stochastic Processes and their Applications, 2012
6. Daniel H., Alexey K., “A Note on the Series Representation for the Density of the Supremum of a Stable Process”, Electron. Commun. Probab., 18 (2013), 1–5
7. V. S. Buyarov, “On the disc of meromorphy of a regular $C$-fraction”, Sb. Math., 206:2 (2015), 201–210
8. Grepstad S., Neumueller M., “Asymptotic Behaviour of the Sudler Product of Sines For Quadratic Irrationals”, J. Math. Anal. Appl., 465:2 (2018), 928–960
9. D. S. Lubinsky, “Exact interpolation, spurious poles, and uniform convergence of multipoint Padé approximants”, Sb. Math., 209:3 (2018), 432–448
10. Lubinsky D.S., “On Uniform Convergence of Diagonal Multipoint Pade Approximants For Entire Functions”, Constr. Approx., 49:1 (2019), 149–174
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