  In a previous paper, we studied an overpartition analogue of Gaussian
polynomials as the generating function for overpartitions fitting inside an $m
\times n$ rectangle. Here, we add one more parameter counting the number of
overlined parts, obtaining a two-parameter generalization $\overline{{m+n
\brack n}}_{q,t}$ of Gaussian polynomials, which is also a $(q,t)$-analogue of
Delannoy numbers. First we obtain finite versions of classical $q$-series
identities such as the $q$-binomial theorem and the Lebesgue identity, as well
as two-variable generalizations of classical identities involving Gaussian
polynomials. Then, by constructing involutions, we obtain an identity involving
a finite theta function and prove the $(q,t)$-log concavity of $\overline{{m+n
\brack n}}_{q,t}$. We particularly emphasize the role of combinatorial proofs
and the consequences of our results on Delannoy numbers. We conclude with some
conjectures about the unimodality of $\overline{{m+n \brack n}}_{q,t}$.
