  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.
