Singular values of the Rogers-Ramanujan continued fraction
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.
scientific article; zbMATH DE number 5070738
scientific article