Congruences for some partitions related to mock theta functions

Abstract

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.

Generating functions and congruences for some partition functions related to mock theta functions

“…There are several congruences for spt ω (n) modulo 11 and powers of 2 and 3 (For example, see [6,28,14]). But for modulo 5, to our knowledge, the following congruence, found by Andrews et al [6], is the only available one: spt ω (10n + 6) ≡ 0 (mod 5).…”

“…They also proved several congruences modulo 2 and infinite families of congruences modulo 4 and modulo 8 for p ω (n) and p ν (n). Motivated by the works in [5,7], Wang [28] and Cui, Gu and Hao [14] found many new congruences satisfied by p ω (n) and p ν (n) modulo 11 and modulo powers of 2 and 3. In particular, Wang [28] derived the following exact generating functions: .…”

“…Recently, the question was answered in affirmative by Bringmann, Jennings-Shaffer and Mahlburg [12]. Recently, Wang [28] and Cui, Gu and Hao [14] also found many new congruences satisfied by p ω (n), spt ω (n) and spt ω (n) modulo powers of 2 and 3. In particular, Wang [28] derived the following exact generating functions:…”

A conjecture of Baruah and Begum on the smallest parts function of restricted overpartitions

“…Moreover, they [2] derived several congruences modulo 2, 3, 4, 5 and 6 for spt ω (n). Soon after, Wang [15] as well as Cui, Gu and Hao [10] also established several new congruences modulo powers of 2 and 3 satisfied by spt ω (n). In particular, Wang [15] proved the following generating function:…”

An infinite family of congruences arising from a second order mock theta function

“…product or quotient of certain q-shifted factorials). The interested readers may refer to [2,6,13] for details. Since the r.h.s.…”