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 $\eta(s)$.