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

This article is cited in 10 scientific papers (total in 10 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

Full text: PDF file (360 kB)
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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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\by V.~I.~Buslaev
\paper Convergence of the Rogers--Ramanujan continued fraction
\jour Mat. Sb.
\yr 2003
\vol 194
\issue 6
\pages 43--66
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\transl
\jour Sb. Math.
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\vol 194
\issue 6
\pages 833--856
\crossref{https://doi.org/10.1070/SM2003v194n06ABEH000741}
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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.  crossref  isi
    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  crossref  mathscinet  zmath  isi  elib
    3. 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
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. A. Kuznetsov, “On the density of the supremum of a stable process”, Stochastic Processes and their Applications, 2012  crossref  mathscinet  isi  scopus
    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  crossref  mathscinet  isi  scopus
    7. 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
    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  crossref  mathscinet  zmath  isi  scopus
    9. D. S. Lubinsky, “Exact interpolation, spurious poles, and uniform convergence of multipoint Padé approximants”, Sb. Math., 209:3 (2018), 432–448  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    10. 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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