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Thursday, March 31, 2016

**Abstract:** Let $\omega(q)$ denote the third order mock theta function of Ramanujan and Watson. Recently George E. Andrews, Ae Ja Yee and I showed that $q\omega(q)$ 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. We also studied the associated smallest parts partition function $\mathrm{spt}_{\omega}(n)$ and proved some congruences for the same. Very recently, we considered the overpartition analogue of $p_{\omega}(n)$, namely, $\overline{p}_{\omega}(n)$. Finding an alternate representation for the generating function of $\overline{p}_{\omega}(n)$ turns out to be difficult in this case. We devise a new seven parameter $q$-series identity which generalizes a deep identity of Andrews (as well as its generalization by R. P. Agarwal), and then specialize it, along with the use of some identities in basic hypergeometric series, to arrive at an alternate representation in terms of a ${}_3\phi_{2}$ and an infinite series involving the little $q$-Jacobi polynomials. We also prove some congruences for $\overline{p}_{\omega}(n)$ and for the overpartition analogue of $\mathrm{spt}_{\omega}(n)$. This is joint work with George E. Andrews, Daniel P. Schultz and Ae Ja Yee.