30 research outputs found
Congruences for (2, 3)-regular partition with designated summands
Let count the number of partitions of with designated summands in which parts are not multiples of or . In this work, we establish congruences modulo powers of 2 and 3 for . For example, for each \quad and \quad 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