J. Korean Math. Soc. 2025; 62(1): 97-126
Online first article December 18, 2024 Printed January 1, 2025
https://doi.org/10.4134/JKMS.j230599
Copyright © The Korean Mathematical Society.
Nayandeep Deka Baruah ; Pranjal Talukdar
Tezpur University; Tezpur University
We prove some new modular identities for the Rogers-\linebreak Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers-Ramanujan continued fraction, then \begin{align*}&R(q)R(q^4)=\dfrac{R(q^5)+R(q^{20})-R(q^5)R(q^{20})}{1+R(q^{5})+R(q^{20})},\\ &\dfrac{1}{R(q^{2})R(q^{3})}+R(q^{2})R(q^{3})= 1+\dfrac{R(q)}{R(q^{6})}+\dfrac{R(q^{6})}{R(q)}, \end{align*} and \begin{align*}&R(q^2)\\ =&\ \dfrac{R(q)R(q^3)}{R(q^6)}\cdot\dfrac{R(q) R^2(q^3) R(q^6)+2 R(q^6) R(q^{12})+ R(q) R(q^3) R^2(q^{12})}{R(q^3) R(q^6)+2 R(q) R^2(q^3) R(q^{12})+ R^2(q^{12})}.\end{align*} In the process, we also find some new relations for the Rogers-Ramanujan functions by using dissections of theta functions and the quintuple product identity.
Keywords: Rogers-Ramanujan functions, Rogers-Ramanujan continued fraction, Ramanujan's notebooks, Ramanujan's lost notebook, modular identities, theta functions
MSC numbers: Primary 11F27, 11P84; Secondary 11A55, 33D90
Supported by: The second author was partially supported by Council of Scientific & Industrial Research (CSIR), Government of India under CSIR-JRF scheme (Grant No. 09/0796(12991)/2021-EMR-I). The author thanks the funding agency.
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