Congruence properties modulo powers of 2 for overpartitions and overpartition pairs

Abstract

In 2004, Corteel and Lovejoy introduced the notion of overpartitions in order to give a combinatorial proof of several celebrated [Formula: see text]-series identities. Let [Formula: see text] denote the number of overpartitions of [Formula: see text]. Many scholars have been investigated subsequently congruence properties modulo powers of [Formula: see text] satisfied by [Formula: see text]. Congruence properties modulo powers of 2 for [Formula: see text] were also considered by several scholars, where [Formula: see text] denotes the number of overpartition pairs of [Formula: see text]. In this paper, utilizing some [Formula: see text]-series identities and iterative computations, we prove several internal congruences and congruences modulo powers of 2 enjoyed by [Formula: see text] and [Formula: see text]. Moreover, we conjecture that these internal congruences and congruences are initial cases in the corresponding internal congruence families and congruence families. Finally, we pose a related conjecture and some questions that merit further investigation.