Overpartitions With Repeated Smallest Non-Overlined Part

Inspired by the recent work of Andrews and El Bachraoui on partitions with repeated smallest part, we extend the concept to overpartitions. We study overpartitions with the restriction that the smallest non-overlined part appears exactly k times and every overlined part is larger than this part. We prove results expressing the generating functions of these overpartitions (and their subclass where no part has the same parity as the smallest part, among others) as linear combinations of the q-Pochhammer symbols with rational functions in q as coefficients, a structure that is new to the overpartition setting. As an application, we obtain identities decomposing several natural overpartition-counting functions into finer subclasses.