Ranks of overpartitions modulo 4 and 8

Abstract

An overpartition of [Formula: see text] is a partition of [Formula: see text] in which the first occurrence of a number may be overlined. Then, the rank of an overpartition is defined as its largest part minus its number of parts. Let [Formula: see text] be the number of overpartitions of [Formula: see text] with rank congruent to [Formula: see text] modulo [Formula: see text]. In this paper, we study the rank differences of overpartitions [Formula: see text] for [Formula: see text] or [Formula: see text] and [Formula: see text]. Especially, we obtain some relations between the generating functions of the rank differences modulo [Formula: see text] and [Formula: see text] and some mock theta functions. Furthermore, we derive some equalities and inequalities on ranks of overpartitions modulo [Formula: see text] and [Formula: see text].