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

Quintuple product identity as a special case of Ramanujan's 1ψ1 summation formula.

In this note we observe an interesting fact that the well-known quintuple product identity can be regarded as a special case of the celebrated 1ψ1 summation formula of Ramanujan which is known to unify the Jacobi triple product identity and the q-binomial theorem

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

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}.

Certain new modular identities for Ramanujan's cubic continued fraction.

In this paper, first we establish some new relations for ratios of Ramanujan’s theta functions. We establish some new general formulas for explicit evaluations of Ramanujan’s theta functions. We also establish new relations connecting Ramanujan’s cubic continued fraction V (q) with four other continued fractions V (q 15), V (q 5/3), V (q 21) and V (q 7/3)

Congruences for Overpartitions with Restricted Odd Differences

In recent work, Bringmann et al. used q-difference equations to compute a two-variable q-hypergeometric generating function for the number of overpartitions where (i) the difference between two successive parts may be odd only if the larger of the two is overlined, and (ii) if the smallest part is odd then it is overlined, given by t ¯ ( n ) . They also established the two-variable generating function for the same overpartitions where (i) consecutive parts differ by a multiple of ( k + 1 ) unless the larger of the two is overlined, and (ii) the smallest part is overlined unless it is divisible by k + 1 , enumerated by t ¯ ( k ) ( n ) . As an application they proved that t ¯ ( n ) = 0 ( mod 3 ) if n is not a square. In this paper, we extend the study of congruence properties of t ¯ ( n ) , and we prove congruences modulo 3 and 6 for t ¯ ( n ) , congruences modulo 2 and 4 for t ¯ ( 3 ) ( n ) and t ¯ ( 7 ) ( n ) , congruences modulo 4 and 5 for t ¯ ( 4 ) ( n ) , and congruences modulo 3, 6 and 12 for t ¯ ( 8 ) ( n )

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 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 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