In a previous paper, we studied an overpartition analogue of Gaussianpolynomials as the generating function for overpartitions fitting inside an $m\times n$ rectangle. Here, we add one more parameter counting the number ofoverlined parts, obtaining a two-parameter generalization $\overline{{m+n\brack n}}_{q,t}$ of Gaussian polynomials, which is also a $(q,t)$-analogue ofDelannoy numbers. First we obtain finite versions of classical $q$-seriesidentities such as the $q$-binomial theorem and the Lebesgue identity, as wellas two-variable generalizations of classical identities involving Gaussianpolynomials. Then, by constructing involutions, we obtain an identity involvinga 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 proofsand the consequences of our results on Delannoy numbers. We conclude with someconjectures about the unimodality of $\overline{{m+n \brack n}}_{q,t}$.
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机译：在先前的论文中，我们研究了高斯多项式的超分区模拟，作为拟合在$ m \ times n $矩形内的超分区的生成函数。在这里，我们再加上一个参数，计算上划线部分的数量，从而获得高斯多项式的两参数概括$ \ overline {{m + n \ brack n}} _ {q，t} $，这也是$（q ，t）$-Delannoy数的类似物。首先，我们获得古典$ q $-系列恒等式的有限形式，例如$ q $-二项式定理和Lebesgue等式，以及涉及高斯多项式的古典恒等式的二变量归纳。然后，通过构造对合，我们获得一个涉及有限theta函数的恒等式，并证明$ \ overline {{m + n \ brack n}} _ {q，t} $的$（q，t）$对数凹度。我们特别强调组合证明的作用以及我们的结果对Delannoy数的影响。我们以一些关于$ \ overline {{m + n \ brack n}} _ {q，t} $的单峰性的猜想来结束。
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