Modulo $d$ extension of parity results in Rogers-Ramanujan-Gordon type overpartition identities

TL;DR

This paper extends overpartition analogs of Rogers-Ramanujan-Gordon identities to arbitrary moduli, unifies various proofs, and addresses a previously open parity-involving question in overpartition identities.

Contribution

It generalizes Sang, Shi, and Yee's work to any modulus, fills a missing case, and unifies multiple existing proofs of overpartition identities.

Findings

Extended identities to arbitrary moduli

Unified proofs of overpartition identities

Provided a construction via functional equations

Abstract

Sang, Shi and Yee, in 2020, found overpartition analogs of Andrews' results involving parity in Rogers-Ramanujan-Gordon identities. Their result partially answered an open question of Andrews'. The open question was to involve parity in overpartition identities. We extend Sang, Shi, and Yee's work to arbitrary moduli, and also provide a missing case in their identities. We also unify proofs of Rogers-Ramanujan-Gordon identities for overpartitions due to Lovejoy and Chen et.al.; Sang, Shi, and Yee's results; and ours. Although verification type proofs are given for brevity, a construction of series as solutions of functional equations between partition generating functions is sketched.