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Sbornik: Mathematics, 2003, Volume 194, Issue 6, Pages 833–856
DOI: https://doi.org/10.1070/SM2003v194n06ABEH000741
(Mi sm741)
 

This article is cited in 13 scientific papers (total in 13 papers)

Convergence of the Rogers–Ramanujan continued fraction

V. I. Buslaev

Steklov Mathematical Institute, Russian Academy of Sciences
References:
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 $\Bigl\{|z|<\max\bigl(R_q,\frac1{2+|1+q|}\bigr)\Bigr\}\setminus\Omega_q$ was established earlier by Lubinsky.
Received: 02.12.2002
Bibliographic databases:
Document Type: Article
UDC: 517.524
MSC: 30B70, 41A21
Language: English
Original paper language: Russian
Citation: V. I. Buslaev, “Convergence of the Rogers–Ramanujan continued fraction”, Sb. Math., 194:6 (2003), 833–856
Citation in format AMSBIB
\Bibitem{Bus03}
\by V.~I.~Buslaev
\paper Convergence of the Rogers--Ramanujan continued fraction
\jour Sb. Math.
\yr 2003
\vol 194
\issue 6
\pages 833--856
\mathnet{http://mi.mathnet.ru//eng/sm741}
\crossref{https://doi.org/10.1070/SM2003v194n06ABEH000741}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1992176}
\zmath{https://zbmath.org/?q=an:1067.30007}
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\elib{https://elibrary.ru/item.asp?id=13419794}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-0142023206}
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  • https://www.mathnet.ru/eng/sm/v194/i6/p43
  • This publication is cited in the following 13 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Математический сборник - 1992–2005 Sbornik: Mathematics
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    References:44
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