Around Poisson-Mehler summation formula

作者：Szablowski Pawel J^*

来源：Hacettepe Journal of Mathematics and Statistics, 2016, 45(6): 1729-1742.

DOI：10.15672/HJMS.20164515997

• q-Hermite
• big q-Hermite
• Al-Salam-Chihara
• orthogonal polynomials
• Poisson-Mehler summation formula
• Orthogonal polynomials on the plane

We study polynomials in x and y of degree n + m : {Q(m,n) (x, y\t,q)}(n,m >= 0) that are related to the generalization of Poisson Mehler formula i.e. to the expansion Sigma(i >= 0) t(i)/[i](q!) Hi+n(x/q) = Q(n,m) (x,y\t,q) Sigma(i >= 0) t(i)/[i](q!) H-i(x/q)(y vertical bar q), where {H-n(x/q)}(n >= 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 >= 0) are orthogonal with respect to a certain measure (two-dimensional (t, q) Normal distribution) on the square {(x, y) :vertical bar x vertical bar,vertical bar y vertical bar <= 2/root 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.

• 出版日期2016-12