### 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 $\eta(s)$.

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 $\eta(s)$.

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