Around Poisson--Mehler summation formula

Abstract

We study polynomials in $x$ and $y$ of degree $n+m:\{Q_{m,n}(x,y|t,q){\}}_{n,m\ge 0}$ that appeared recently in the following identity: ${\gamma }_{m,n}(x,y|t,q)={\gamma }_{0,0}(x,y|t,q)Q_{m,n}(x,y|t,q)$ where ${\gamma }_{m,n}(x,y|t,q)={\sum }_{i\ge 0}\frac{t^i}{[i{]}_q}{H}_{i+n}(x|q)H_{m+i}(y|q)$, $\{H_{n}(x|q)}_{n\geq -1}$ are the so-called $q-$% Hermite polynomials (qH). In particular we show that the spaces $span\{Q_{i,n-i}(x,y|t,q):i=0,...,n{\}}_{n\ge 0}$ are orthogonal with respect to a certain measure (two-dimensional $(t,q)-$Normal distribution) on the square $\{(x,y):|x|,|y|\le 2/\sqrt{1-q}\}.$ 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.

Cited by in corpus (2)

• On three dimensional multivariate version of q-Normal distribution and probabilistic interpretations of Askey--Wilson, Al-Salam--Chihara and q-ultraspherical polynomials
• On the generalized Kesten--McKay distributions