Certain eta-quotients and \(\ell \)-regular partitions with distinct odd parts

Let \(\text {pod}_{\ell }(n)\) ${\text{pod}}_{\ell }(n)$ be the number of \(\ell \) $\ell$ -regular partitions of n with distinct odd parts. In this article, we prove that for any positive integer k, the set of non-negative integers n for which \(\text {pod}_{\ell }(n)\equiv 0 \pmod {p^{k}}\) ${\text{pod}}_{\ell }(n)\equiv 0(mod{p}^k)$ has density one. We also exhibit several multiplicative identities for \(\text {pod}_{3}(n)\) ${\text{pod}}_3(n)$ , \(\text {pod}_{5}(n)\) ${\text{pod}}_5(n)$ and \(\text {pod}_{7}(n)\) ${\text{pod}}_7(n)$ using the Hecke eigenforms, and some results of Ono, Robins, and Wahl.