Around Poisson--Mehler summation formula

Details Around Poisson--Mehler summation formula Around Poisson--Mehler summation formula

2011-08-15

arxiv.org/abs/1108.3024v2

Mathematics

Classical Analysis and ODEs

Scientific paper

We study some simple generalization of the Poisson-Mehler summation formula (PM). Namely we exploit farther, the recently obtained equality {\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\geq0}((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) and {Q_{m,n}(x,y|t,q)}_{n,m\geq0} are certain polynomials in x,y of order m+n that are also rational functions in t and q. We study properties of polynomials Q_{m,n}(x,y|t,q) expressing them with the help the so called Al-Salam-Chihara (ASC) polynomials and using them in expansion of the reciprocal of the right hand side of the Poisson-Mehler formula. We also point out the fact that spaces span{Q_{i,n-i}(x,y|t,q):i=0,...,n}_{n\geq0} are orthogonal with respect to certain two dimensional measure (two dimensional (t,q)-Normal distribution) on the square {(x,y):|x|,|y|\leq2/\surd(1-q)}.

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