A continued fraction of Ramanujan and some Ramanujan-Weber class invariants

Abstract

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 $G_n$. We obtain relations between our continued fraction with the Ramanujan-G\{o}llnitz-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-G\{o}llnitz-Gordon continued fraction $H(q)$ and the Dedekind eta function $\eta(s)$.