Overpartitions with repeated smallest non-overlined part

TL;DR

This paper extends the concept of partitions with repeated smallest parts to overpartitions, deriving generating functions for these structures with specific restrictions.

Contribution

It introduces a new class of overpartitions with restrictions on the smallest non-overlined part and derives their generating functions in terms of q-Pochhammer symbols.

Findings

Generated explicit formulas for overpartition generating functions.

Extended existing partition theories to overpartitions with new restrictions.

Provided algebraic expressions involving rational functions in q.

Abstract

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.