On partitions with bounded multiplicity restrictions on even parts and with the largest part being odd

Andrews and El Bachraoui (\textit{RACSAM} \textbf{119}, \textit{Article Number 4 (2025)}) established identities relating $4$-regular partitions to partitions with distinct even parts whose largest part is odd. As an extension to their work, in this paper we define partitions of a positive integer $n$ with bounded multiplicity restriction on even parts and whose largest part is odd. We prove some identities relating our partition functions and $2r$-regular partition function, and show that identities due to Andrews and El Bachraoui follow as particular cases.