## Around Poisson--Mehler summation formula

### Paweł Szabłowski

#### Abstract

We study polynomials in $x$ and $y$ of degree $n+m:allowbreak$ ewline% $left{ Q_{m,n}left( x,y|t,q ight) ight} _{n,mgeq0}$ that are related to the generalization of Poisson--Mehler formula i.e. to the expansion $allowbreaksum_{igeq0}frac{t^{i}}{left[ i ight] _{q}!}H_{i+n}left( x|q ight) H_{m+i}(y|q)$ ewline$=Q_{n,m}(x,y|t,q)sum_{igeq0}frac{t^{i}% }{left[ i ight] _{q}!}H_{i}left( x|q ight) H_{m}(y|q)$, $allowbreak$where ewline$left{ H_{n}left( x|q ight) ight} _{ngeq-1}$ are the so-called $q-$Hermite polynomials (qH). In particular we show that the spaces $spanleft{ Q_{i,n-i}left( x,y|t,q ight) :i=0,ldots,n ight} _{ngeq0}$ are orthogonal with respect to a certain measure (two-dimensional $(t,q)-$Normal distribution) on the square $left{ (x,y):|x|,|y|leq 2/sqrt{1-q} ight}$ 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.
 Author Paweł Szabłowski (FMIS / DAST) Paweł Szabłowski,, - Department of Analysis and Sigularity Theory Journal series Hacettepe Journal of Mathematics and Statistics, ISSN 1303-5010 Issue year 2016 Vol 45 No 6 Pages 1729-1742 Publication size in sheets 0.65 Keywords in English $q-$Hermite, big $q-$Hermite, Al-Salam--Chihara, orthogonal polynomials, Poisson-Mehler summation formula. Orthogonal polynomials on the plane Abstract in Polish Wychodząc z uogólnienia formuły Poissona--Mehlera dokonanej przez autora w poprzednich publikacjach a mającej formę: $allowbreaksum_igeq0fract^ileft[ i ight] _q!H_i+nleft( x|q ight) H_m+i(y|q)$ ewline$=Q_n,m(x,y|t,q)sum_igeq0fract^i% left[ i ight] _q!H_ileft( x|q ight) H_m(y|q)$, $allowbreak$where ewline$left H_nleft( x|q ight) ight _ngeq-1$ w pracy analizuje się formę i rolę wielomianów Q_n,m(x,y|t,q). Zależą one od 2- zmiennych x i y i są stopnia n+m. Okazuje się że są one ortogonalne, jeśli tylko sumy n+m są różne, względem miary będącej uogólnieniem 2 wymiarowego normalnego. Ponadto wskazano rozwinięcia pewnych funkcji analitycznych w szeregi wielomianów Q_n,m(x,y|t,q). DOI DOI:10.15672/HJMS.20164515997 URL http://www.hjms.hacettepe.edu.tr/uploads/7faa0da3-97ba-43b7-8a98-804fa2fc84fa.pdf Language en angielski Score (nominal) 20 Score Ministerial score = 15.0, 28-11-2017, ArticleFromJournalMinisterial score (2013-2016) = 20.0, 28-11-2017, ArticleFromJournal Publication indicators WoS Impact Factor: 2016 = 0.415 (2) - 2016=0.541 (5) Citation count*
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