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Partitions related to mock theta functions were widely studied in the literature. Recently, Andrews et al. introduced two new kinds of partitions counted by [Formula: see text] and [Formula: see text], whose generating functions are [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are two third mock theta functions. Meanwhile, they obtained some congruences for [Formula: see text], [Formula: see text], and the associated smallest parts function [Formula: see text]. Furthermore, Andrews et al. discussed the overpartition analogues of [Formula: see text] and [Formula: see text] which are denoted by [Formula: see text] and [Formula: see text]. In this paper, we derive more congruences for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Moreover, we establish some congruences for [Formula: see text] and its associated smallest parts function [Formula: see text], where [Formula: see text] denotes the number of overpartitions of [Formula: see text] such that all even parts are at most twice the smallest part, and in which the smallest part is always overlined.

In 1991, Andrews and Hickerson established a new Bailey pair and combined it with the constant term method to prove some results related to sixth-order mock theta functions. In this paper, we study how this pair gives rise to new mock theta functions in terms of Appell–Lerch sums. Furthermore, we establish some relations between these new mock theta functions and some second-order mock theta functions. Meanwhile, we obtain an identity between a second-order and a sixth-order mock theta functions. In addition, we provide the mock theta conjectures for these new mock theta functions. Finally, we discuss the dual nature between the new mock theta functions and partial theta functions.

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