  We obtain a three-parameter $q$-series identity that generalizes two results
of Chan and Mao. By specializing our identity, we derive new results of
combinatorial significance in connection with $N(r, s, m, n)$, a function
counting certain overpartition pairs recently introduced by Bringmann, Lovejoy
and Osburn. For example, one of our identities gives a closed-form evaluation
of a double series in terms of Chebyshev polynomials of the second kind,
thereby resulting in an analogue of Euler's pentagonal number theorem. Another
of our results expresses a multi-sum involving $N(r, s, m, n)$ in terms of just
the partition function $p(n)$. Using a result of Shimura we also relate a
certain double series with a weight 7/2 theta series.
