Some theorems on the explicit evaluations of singular moduli

At scattered places in his notebooks, Ramanujan recorded some theorems for calculating singular moduli and also recorded several values of singular moduli. In this paper, we establish several general theorems for the explicit evaluations of singular moduli. we also obtain some values of class invariants and singular moduli

On some explicit evaluations of the ratios of Ramanujan's theta-function.

In this paper, we establish several new modular equations of degree 9 using Ramanujan’s modular equations. We also establish several general formulas for explicit evaluations of h9,n, h0 9,n, l9,n and l 0 9,n. As an application, we establish some explicit evaluations for Ramanujan’s cubic continued fraction

New identities for Ramanujan's cubic continued fraction.

In this paper, we present some new identities providing relations between Ramanujan's cubic continued fraction V(q)V(q) and the other three continued fractions V(q9)V(q9), V(q17)V(q17) and V(q19)V(q19). In the process, we establish some new modular equations for the ratios of Ramanujan's theta functions. We also establish some general formulas for the explicit evaluations of ratios of Ramanujan's theta functions

On some New Identities for Ramanujan'scubic Continued Fraction

In this paper, we establish some new modular relations connecting Ramanujan's cubic continued fraction V (q) with V (qn), for n = 4, 6, 8,10, 12, 14, 16 and 22

On some new identities for Ramanujan's cubic continued fraction.

In this paper, we establish some new modular relations connecting Ramanujan's cubic continued fraction V (q) with V(qn), for n= 4, 6, 8,10, 12, 14, 16 and 22

Ratios of Ramanujan's Theta Function ψ and Evaluations

In this paper, we establish several new modular equations of degree 9 using Ramanujan's mixed modular equations. We also establish several general formulas for explicit evaluations of ratios of Ramanujan's theta function

Some new identities for a continued fraction of order 12.

In this paper, we establish some new modular equations of degree three. We establish some new relations for continued fraction of order 12 and also for a parameter µ(q) = 2V (q)V (q 2 ) connecting Ramanujan’s cubic continued fraction

On some new Schläfli-type mixed modular equations

On pages 86 and 88 of his first notebook, Ramanujan recorded eleven Schl\"{a}fli-type modular equations for composite degrees. Out of eleven, ten have been proved by Berndt using theory of modular forms. In this paper, we establish several new Schl\"{a}fli-type mixed modular equations

A continued fraction of order 4 found in Ramanujan's `lost' notebook.

On page 44 of his lost notebook, Ramanujan has recorded many continued fractions of orders 4, 5, 6 and 8. In this paper, we establish several interesting results of a continued fraction of order 4 which are analogous to Rogers-Ramanujan and cubic continued fractions

Certain quotient of eta-function identities.

On page 212 in his lost notebook, Ramanujan defined a parameter $\lambda_n$ by a certain quotient of Dedekind eta-functions at the argument $q=exp(-\pi \sqrt{n/3})$. He then recorded a table of several values of $\lambda_n:=\lambda_{n,\,\, 3}$. All these have been established by B. C. Berndt, H. H. Chan, S.-Y. Kang and L.-C. Zhang \cite{BCB4}. In this paper following Ramanujan we defined a parameter $\lambda_{n, p}$ at the argument $q=exp(-\pi \sqrt{n/p})$. We establish several interesting and new explicit evaluations for $\lambda_{n,\,\, p}$ using Ramanujan-Weber class invariant, modular equations and mixed-modular equations