Around Poisson-Mehler summation formula

Authors : Pawel J. SZABLOWSKİ

Pages : 1729-1742

View : 15 | Download : 6

Publication Date : 2016-12-01

Article Type : Research Paper

Abstract :We study polynomials in $x$ and $y$ of degree $n+m$: $\{Q_{m,n}insert ignore into journalissuearticles values(x,y|t,q);\}_{n,m\geq0}$ that are related to the generalization of Poisson-Mehler formula i.e. to the expansion $\sum_{i\geq0}\dfrac{t^{i}}{[i]_q!}H_{i+n}insert ignore into journalissuearticles values(x|q);H_{m+i}insert ignore into journalissuearticles values(y|q);=Q_{n,m}insert ignore into journalissuearticles values(x,y|t,q);\sum_{i\geq0}H_iinsert ignore into journalissuearticles values(x|q);H_minsert ignore into journalissuearticles values(y|q);$, where $\{H_ninsert ignore into journalissuearticles values(x|q);\}_{n\geq-1}$ are the so-called $q-$Hermite polynomials insert ignore into journalissuearticles values(qH);. In particular we show that the spaces $span\{Q_{i,n-i}insert ignore into journalissuearticles values(x,y|t,q);:i=0,\cdots,n\}_{n\geq0}$ are orthogonal with respect to a certain measure insert ignore into journalissuearticles values(two dimensional $insert ignore into journalissuearticles values(t,q);-$Normal distribution); on the square $\{insert ignore into journalissuearticles values(x,y);:|x|,|y|\leq2/\sqrt{1-q}\}$ 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. 
Keywords : q Hermite, big q Hermite, Al Salam Chihara, orthogonal polynomials, Poisson Mehler summation formula, Orthogonal polynomials on the plane