FILOMAT, vol.31, no.13, pp.3975-3997, 2017 (SCI-Expanded)
On Page 36 of his "lost" notebook, Ramanujan recorded four q-series representations of the famous Rogers-Ramanujan continued fraction. In this paper, we establish two q-series representations of Ramanujan's continued fraction found in his "lost" notebook. We also establish three equivalent integral representations and modular equations for a special case of this continued fraction. Furthermore, we derive continued-fraction representations for the Ramanujan-Weber class invariants g(n) and G(n) and establish formulas connecting g(n) and Gn. We obtain relations between our continued fraction with the Ramanujan-Gollnitz-Gordon and Ramanujan's cubic continued fractions. Finally, we find some algebraic numbers and transcendental numbers associated with a certain continued fraction A(q) which is related to Ramanujan's continued fraction F(a; b; lambda; q); the Ramanujan-Gollnitz-Gordon continued fraction H(q) and the Dedekind eta function eta(s).