Congruences for (2, 3)-regular partition with designated summands

Let $PD_{2, 3}(n)$ count the number of partitions of $n$ with designated summands in which parts are not multiples of $2$ or $3$. In this work, we establish congruences modulo powers of 2 and 3 for $PD_{2, 3}(n)$. For example, for each \quad $n\ge0$ and $\alpha\geq0$ \quad $PD_{2, 3}(6\cdot4^{\alpha+2}n+5\cdot4^{\alpha+2})\equiv 0 \pmod{2^4}$ and $PD_{2, 3}(4\cdot3^{\alpha+3}n+10\cdot3^{\alpha+2})\equiv 0 \pmod{3}.

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

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

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