Abstract:
It was recently shown that qω(q), where ω(q) is one of the third order mock theta functions, is the generating function of pω(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 pω(n), and express its generating function in terms of a 3ϕ2 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.