Distribution of ℓ-regular partitions with distinct odd parts

Let $pod_{\ell}(n)$ be the number of $\ell$-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 $pod_{\ell}(n)\equiv 0 \pmod{p^{k}}$ has density one. We also exhibit several multiplicative identities for $pod_{3}(n)$, $pod_{5}(n)$ and $pod_{7}(n)$ using the Hecke eigenforms, and some results of Ono, Robins, and Wahl. 2010 Mathematics Subject Classification. Primary: 05A17, 11P83, 11F11, 11F20