Singular values of the Rogers-Ramanujan continued fraction
type: Item
instance of: scholarly article
title: Singular values of the Rogers-Ramanujan continued fraction
zbMATH Open document ID: 1112.11058
author: Alice Gee
author: Mascha Honsbeek
published in: The Ramanujan Journal
publication date: 2006-11-03T00:00:00Z
review text: In his first letter to G.H. Hardy, S. Ramanujan recorded several identities associated with the Rogers-Ramanujan continued fraction \[ \begin{aligned} R(q) &= \frac{q^{1/5} }{ 1+ \frac{ q}{1+ \frac{q^2}{ 1+ \cdots}}}, \end{aligned} \] one of which is \[ R(e^{-2\pi}) = \sqrt{\frac{5+\sqrt{5}}{2}}-\frac{\sqrt{5}+1}{2}. \] On page 210 of his Lost Notebook, S. Ramanujan indicated that he was able to compute \(R(e^{-2\pi\sqrt{n/5}})\) for any positive integer \(n\). Ramanujan, however, did not provide any further details on his method. Attempts to compute \(R(e^{-2\pi\sqrt{n/5}})\) explicitly were previously made by K. G. Ramanathan, B. C. Berndt, H. H. Chan and L. C. Zhang. In this article, the authors succeeded in finding a method that can be used to express \(R(e^{-2\pi\sqrt{n/5}})\) in terms of nested radicals. They showed that if \( q=e^{2\pi i z} \) with \( z\in \mathbb H\), where \(\mathbb H\) is the complex upper plane, then \(R(q)\) is modular of level 5. Using theorems of complex multiplication, they showed that \(R(q)\) at \(z=\sqrt{-n}, n\in \mathbb Q^+\), generated certain ray class field over \(\mathbb Q(\sqrt{-n})\). By determining the action of the corresponding Galois group associated with the ray class fields on \(R(e^{-2\pi \sqrt{n/5}})\), they were able to evaluate many new singular values of \(R(q)\). This is a very interesting article and it is highly recommended to those who wish to know more about the applications of modular functions in the study of Ramanujan's identities.
reviewed by: Heng Huat Chan
Mathematics Subject Classification ID: 11Y65
Mathematics Subject Classification ID: 11Y40
zbMATH DE Number: 5070738
zbMATH Keywords: Rogers-Ramanujan continued fraction
zbMATH Keywords: Shimura reciprocity
MaRDI profile type: Publication
full work available at URL: s11139-006-8477-7
OpenAlex ID: W2012056599
cites work: Q4716808
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cites work: Ramanujan's formulas for the explicit evaluation of the Rogers-Ramanujan continued fraction and theta-functions
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DOI: 10.1007/S11139-006-8477-7
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description: scientific article; zbMATH DE number 5070738