Around Poisson--Mehler summation formula (1108.3024v4)

Abstract: 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 $span{Q{i,n-i}(x,y|t,q) :i=0,...,n}_{n\geq 0}$ are orthogonal with respect to a certain measure (two-dimensional $(t,q)-$Normal distribution) on the square ${(x,y):|x|,|y|\leq 2/\sqrt{1-q}} . $ 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.