Parity considerations in Rogers–Ramanujan–Gordon type overpartitions

In 2010, Andrews investigated a variety of parity questions in the classical partition identities of Euler, Rogers, Ramanujan and Gordon. In particular, he considered the Rogers-Ramanujan-Gordon partitions with some constraints on even and odd parts. At the end of this paper, he left fifteen open questions, of which the eleventh is to extend his parity consideration to overpartitions. The main purpose of this paper is to undertake that question. In 2013, Chen, Sang and Shi derived an overpartition analogue of the Rogers–Ramanujan–Gordon theorem. Motivated by their work, we define two kinds of Rogers–Ramanujan–Gordon type overpartitions with some parity constraints on even and odd parts. We then provide the generating functions for such partitions in some cases.