Around Poisson--Mehler summation formula

We study polynomials in $x$ and $y$ of degree $n+m:\allowbreak {Q_{m,n}(x,y|t,q)}{n,m\geq 0}$ that appeared recently in the following identity: $\gamma{m,n}(x,y|t,q) \allowbreak =\allowbreak \gamma_{0,0}(x,y|t,q) \allowbreak Q_{m,n}(x,y|t,q) $where$\gamma_{m,n}(x,y|t,q) \allowbreak =\allowbreak \sum_{i\geq 0}\frac{t^{i}}{[i]{q}}H{i+n}(x|q) H_{m+i}(y|q)$,$\allowbreak $ \{H_{n}(x|q)}_{n\geq -1} are the so-called $q-$% Hermite polynomials (qH). In particular we show that the spaces $span\{Q_{i,n-i}(x,y\mathrm{\mid }t,q):i=0,\mathrm{.}\mathrm{.}\mathrm{.},n{\}}_{n\ge 0}$ are orthogonal with respect to a certain measure (two-dimensional $(t,q)-$Normal distribution) on the square $\{(x,y):\mathrm{\mid }x\mathrm{\mid },\mathrm{\mid }y\mathrm{\mid }\le 2\mathrm{/}\sqrt{1-q}\}\mathrm{.}$ We study structure of these polynomials expressing them with the help of the so-called Al-Salam–Chihara (ASC) polynomials and showing that they are rational functions of parameters $t$ and $q$. We use them in various infinite expansions that can be viewed as simple generalization of the Poisson-Mehler summation formula. Further we use them in the expansion of the reciprocal of the right hand side of the Poisson-Mehler formula.