Some New Modular Equations of Degree 2 Akin to Ramanujan

In this paper, we obtain some new modular equations of degree 2 for the ratios of Ramanujan's theta-function f and also establish the general formulas for their explicit evaluations. As an application, we establish some new modular relations for Ramanujan-Göllnitz-Gordon continued fraction H(q) with H(qn/2), Ramanujan-Selberg continued fraction V(q) with V(qn/2) and Eisenstein continued fraction E(q) with E(qn/2) for n=3, 5 and 7

New identities for ratios of ramanujan's theta function

Ramanujan in his notebooks, has established several new modular equation which he denoted as P and Q. In this paper, we establish several new identities for ratios of Ramanujan's theta function involving �(q). We establish some new explicit evaluations for the ratios of Ramanujan's theta function. We also establish some new modular relations for a continued fraction of order twelve II(q) with H(qn) for n =2, 4, 6, 8, 10. 12. 14 and 16

On a new parameter involving Ramanujan's theta-functions

We define a new parameter $A'_{k,n}$ involving Ramanujan's theta-functions for any positive real numbers $k$ and $n$ which is analogous to the parameter $A_{k,n}$ defined by Nipen Saikia \cite{NS1}. We establish some modular relation involving $A'_{k,n}$ and $A_{k,n}$ to find some explicit values of $A'_{k,n}$. We use these parameters to establish few general theorems for explicit evaluations of ratios of theta functions involving $\varphi(q)$

On some New Modular Equations and their Applications to Continued Fractions

In this paper, we obtain some new modular equations of degree2. We obtain several general formulas for the explicit evaluations of the Ramanujan's theta{function. As an application, we establish somenew modular relations for Ramanujan{Gollnitz{Gordon continued frac-tion H(q) with H(qn), Ramanujan{Selberg continued fraction V (q) with V (qn) and Eisenstein continued fraction E(q) with E(qn) for n =6; 10; 14 and 16. We also establish their explicit evaluations

On some new mixed modular equations involving Ramanujan's theta-functions

In his second notebook, Ramanujan recorded altogether 23 P–Q modular equations involving his theta functions. In this paper, we establish several new mixed modular equations involving Ramanujan’s theta-functions ϕ and ψ which are akin to those recorded in his notebook

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

On some new modular equations of degree 9 and their applications

In this paper, we establish several new modular equations of degree 9 using Ramanujan's modular equations. We also establish several new general formulas to compute the values for r 9,n and râ² 9. As an application, we establish explicit evaluations of Ramanujan's remarkable product of theta-functions

Schläfli-type mixed modular equations of degrees 1, 3, n and 3n.

In this paper, we establish several new Schlafli-type mixed modular equations of composite degrees. These equations are analogous to those recorded by Ramanujan in his second notebook. As an application, we establish several new explicit values for the Ramanujan-Weber class invariant Gn for n = 12,48,51,57,3/4,3/16,3/17 and 3/19