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

This article is cited in 9 scientific papers (total in 9 papers)

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
Received: 02.12.2002

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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\paper Convergence of the Rogers--Ramanujan continued fraction
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\pages 43--66
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\transl
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Lukashov A.L., Peherstorfer F., “Zeros of polynomials orthogonal on two arcs of the unit circle”, J. Approx. Theory, 132:1 (2005), 42–71  crossref  mathscinet  zmath  isi  elib
    2. Buslaev V.I., “On Hankel Determinants of Functions Given by their Expansions in P-Fractions”, Ukrainian Mathematical Journal, 62:3 (2010), 358–372  crossref  mathscinet  zmath  isi  elib
    3. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. A. Kuznetsov, “On the density of the supremum of a stable process”, Stochastic Processes and their Applications, 2012  crossref  mathscinet  isi  scopus
    5. 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  crossref  mathscinet  isi  scopus
    6. V. S. Buyarov, “On the disc of meromorphy of a regular $C$-fraction”, Sb. Math., 206:2 (2015), 201–210  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. Grepstad S., Neumueller M., “Asymptotic Behaviour of the Sudler Product of Sines For Quadratic Irrationals”, J. Math. Anal. Appl., 465:2 (2018), 928–960  crossref  mathscinet  zmath  isi  scopus
    8. D. S. Lubinsky, “Exact interpolation, spurious poles, and uniform convergence of multipoint Padé approximants”, Sb. Math., 209:3 (2018), 432–448  mathnet  crossref  crossref  adsnasa  isi  elib
    9. Lubinsky D.S., “On Uniform Convergence of Diagonal Multipoint Pade Approximants For Entire Functions”, Constr. Approx., 49:1 (2019), 149–174  crossref  mathscinet  zmath  isi  scopus
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