Around Poisson--Mehler summation formula

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We study polynomials in x and y of degree n+m: {Qₘ,ₙ(x,y|t,q)}ₙ,ₘ≥ ₀ that appeared recently in the following identity: γₘ,ₙ(x,y|t,q) = γ₀,₀(x,y|t,q) Qₘ,ₙ(x,y|t,q) where γₘ,ₙ(x,y|t,q) = ∑ᵢ≥ ₀tⁱ[i]qHᵢ₊ₙ(x|q) Hₘ₊ᵢ(y|q), {Hₙ(x|q)ₙ≥ ₋₁ are the so-called q-% Hermite polynomials (qH). In particular we show that the spaces span{Qᵢ,ₙ₋ᵢ(x,y|t,q):i=0,...,n}ₙ≥ ₀ are orthogonal with respect to a certain measure (two-dimensional (t,q)-Normal distribution) on the square {(x,y):|x|,|y|≤ 2/√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.