Around Poisson-Mehler summation formula

Abstract

We study polynomials in $x$ and $y$ of degree $n+m$: $\{Q_{m,n}(x,y|t,q)\}_{n,m\geq0}$ that are related to the generalization of Poisson-Mehler formula i.e. to the expansion $\sum_{i\geq0}\dfrac{t^{i}}{[i]_q!}H_{i+n}(x|q)H_{m+i}(y|q)=Q_{n,m}(x,y|t,q)\sum_{i\geq0}H_i(x|q)H_m(y|q)$, where $\{H_n(x|q)\}_{n\geq-1}$ are the so-called $q-$Hermite polynomials (qH). In particular we show that the spaces $span\{Q_{i,n-i}(x,y|t,q):i=0,\cdots,n\}_{n\geq0}$ are orthogonal with respect to a certain measure (two dimensional $(t,q)-$Normal distribution) on the square $\{(x,y):|x|,|y|\leq2/\sqrt{1-q}\}$ being a generalization of two-dimensional Gaussian measure. We study structure of these polynomials showing in particular 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.