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

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 ${}_3{\varphi }_2$ 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.

25 pages, submitted for publication