Overpartitions with repeated smallest non-overlined part

Inspired by Andrews’ and Bachraoui’s work 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 bigger 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.