Modular equations for the Rogers-Ramanujan continued fraction and the Dedekind eta-function
For \(| q|<1\), let \(R(q)\) denote the Ramanujan continued fraction, that is, \[ R(q)=\frac {q^{1/5}}{1+ \frac {q}{1+ \frac {q^2}{1+ \frac {q^3}{\ddots}}}} \] In his Lost Notebook, S. Ramanujan found five stunning polynomial identities relating \(R(q)\) and \(R(-q)\), \(R(q^2)\), \(R(q^3)\), \(R(q^4)\), and \(R(q^5)\), respectively. The author provides new proofs for these identities and, in addition, proves a new relation between \(u=R(q)\) and \(v=R(q^7)\), namely \[ \frac {1} {7}(u^7-v)(1+uv^7)=uv^3(1-u^5)+u^2v^5(1+u^5) -u^5v(1-v^5)-u^3v^2(1+v^5)-5u^4v^4. \] The proofs are based on certain modular equations, which are established by the use of the theory of modular forms.
scientific article; zbMATH DE number 1738885