Parity Considerations in Rogers-Ramanujan-Gordon Type Overpartitions

In 2010, Andrews considers a variety of parity questions connected to classical partition identities of Euler, Rogers, Ramanujan and Gordon. As a large part in his paper, Andrews considered the partitions by restricting the parity of occurrences of even numbers or odd numbers in the Rogers-Ramanujan-Gordon type. The Rogers-Ramanujan-Gordon type partition was defined by Gordon in 1961 as a combinatorial generalization of the Rogers-Ramaujan identities with odd moduli. In 1974, Andrews derived an identity which can be considered as the generating function counterpart of the Rogers-Ramanujan-Gordon theorem, and since then it has been called the Andrews--Gordon identity. By revisting the Andrews--Gordon identity Andrews extended his results by considering some additional restrictions involving parities to obtain some Rogers-Ramanujan-Gordon type theorems and Andrews--Gordon type identities. In the end of Andrews' paper, he posed \(15\) open problems. Most of Andrews' \(15\) open problems have been settled, but the \(11\)th that "extend the parity indices to overpartitions in a manner" has not. In 2013, Chen, Sang and Shi, derived the overpartition analogues of the Rogers-Ramanujan-Gordon theorem and the Andrews-Gordon identity. In this paper, we post some parity restrictions on these overpartitions analogues to get some Rogers-Ramanujan-Gordon type overpartition theorems.