Overpartitions related to the mock theta function $\omega(q)$

TL;DR

This paper explores the overpartition analogue of a mock theta function related to partition counts, expressing its generating function with hypergeometric series, deriving new identities, and establishing interesting congruences.

Contribution

It introduces a new seven parameter q-series identity generalizing previous identities and connects overpartition functions to hypergeometric series and orthogonal polynomials.

Findings

Derived a new seven parameter q-series identity.

Expressed the overpartition generating function using hypergeometric series.

Established novel congruences for the overpartition analogue and smallest parts function.

Abstract

It was recently shown that $q\omega(q)$, where $\omega(q)$ is one of the third order mock theta functions, is the generating function of $p_{\omega}(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less than twice the smallest part. In this paper, we study the overpartition analogue of $p_{\omega}(n)$, and express its generating function in terms of a ${}_3\phi_{2}$ basic hypergeometric series and an infinite series involving little $q$-Jacobi polynomials. This is accomplished by obtaining a new seven parameter $q$-series identity which generalizes a deep identity due to the first author as well as its generalization by R.P.~Agarwal. We also derive two interesting congruences satisfied by the overpartition analogue, and some congruences satisfied by the associated smallest parts function.