Let pod 2 (n) denote the number of 2-regular partitions of n with distinct odd parts (even parts are unrestricted). In this article, we obtain congruences for pod 2 (n) mod 2 and mod 8 using some generating function manipulations and the theory of Hecke eigenform.

We establish several partition identities with distinct colors that arise from Ramanujan's theta-function identities and formulas for multipliers in the theory of modular equations. Also, we deduce few partition congruences as a corollary of some partition identities.

In this paper, we establish some new modular equations of degree 9. We also establish several new P -Q mixed modular equations involving theta-functions which are similar to those recorded by Ramanujan in his notebooks. As an application, we establish some new general formulas for explicit evaluations of a Remarkable product of theta-functions.

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