[Paper Review] Around Poisson--Mehler summation formula

Paweł J. Szabłowski|arXiv (Cornell University)|Aug 15, 2011

Mathematical functions and polynomials3 citations

TL;DR

This paper introduces a two-variable polynomial family $ Q_{m,n}(x,y|t,q) $ that generalizes the Poisson-Mehler summation formula via $ q $-Hermite and Al-Salam–Chihara polynomials. It establishes that these polynomials form orthogonal bases for a two-dimensional $(t,q)$-Normal distribution, enabling symmetric expansions and reciprocal series representations, with key results on generating functions and orthogonal decomposition on a bounded square domain.

ABSTRACT

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

Motivation & Objective

• To generalize the Poisson-Mehler summation formula using $ q $-Hermite polynomials and introduce a new class of bivariate orthogonal polynomials $ Q_{m,n}(x,y|t,q) $.
• To characterize the orthogonal structure of $ Q_{m,n} $ with respect to a two-dimensional $(t,q)$-Normal measure on the square $ |x|,|y| \leq 2/\sqrt{1-q} $.
• To derive generating functions and symmetric infinite expansions involving $ Q_{m,n} $, including a symmetric expansion of the reciprocal of the Poisson-Mehler kernel.
• To express $ Q_{m,n} $ in terms of Al-Salam–Chihara polynomials and show they are rational functions of $ t $ and $ q $.
• To establish that the spaces $ \Lambda_n(x,y|t,q) = \text{span}\{ Q_{i,n-i}(x,y|t,q) \}_{i=0}^n $ form an orthogonal decomposition of $ L^2 $ under the $(t,q)$-Normal measure.

Proposed method

• Define $ \gamma_{m,n}(x,y|t,q) = \sum_{i \geq 0} \frac{t^i}{[i]_q!} H_{i+n}(x|q) H_{m+i}(y|q) $, and factor it as $ \gamma_{m,n} = Q_{m,n} \cdot \gamma_{0,0} $, establishing the polynomial nature of $ Q_{m,n} $.
• Use the Poisson-Mehler formula as a foundation, extending it via $ q $-series identities and $ q $-exponential generating functions involving $ q $-Hermite and Al-Salam–Chihara polynomials.
• Construct the $(t,q)$-Normal measure on $ S(q) \times S(q) $, where $ S(q) = \{ x : |x| \leq 2/\sqrt{1-q} \} $, using products of $ q $-Hermite and Al-Salam–Chihara orthogonal measures.
• Derive the generating function $ \sum_{i,j \geq 0} \frac{s^i t^j}{[i]_q! [j]_q!} Q_{i,j}(x,y|t,q) = \frac{\varphi_H(x|t,q) \varphi_H(y|s,q)}{\gamma_{0,0}(x,y|t,q)} \sum_{n \geq 0} \frac{t^n}{[n]_q!} H_n(x|t,q) H_n(y|s,q) $, linking to the Poisson-Mehler kernel.
• Prove that $ \Lambda_n(x,y|t,q) = \text{span}\{ Q_{i,n-i}(x,y|t,q) \}_{i=0}^n $ are orthogonal subspaces under the $(t,q)$-Normal measure, forming a direct sum decomposition of $ L^2 $.
• Use induction and $ q $-binomial identities to derive symmetric identities, including $ \sum_{k=0}^n (-1)^k q^{\binom{n-k}{2}} \binom{n}{k}_q \frac{\prod_{i=0}^{k-1} \omega(\cdot | \rho q^i)}{(\rho^2)_{2k}} = q^{\binom{n}{2}} \rho^n (1-q)^n Q_{n,n}(x,y|\rho,q) $.

Experimental results

Research questions

• RQ1 How can the Poisson-Mehler summation formula be generalized to bivariate $ q $-Hermite polynomials using a new class of orthogonal polynomials $ Q_{m,n}(x,y|t,q) $?
• RQ2 What is the orthogonal structure of the spaces $ \Lambda_n(x,y|t,q) $, and how do they decompose $ L^2 $ under the $(t,q)$-Normal measure?
• RQ3 Can the reciprocal of the Poisson-Mehler kernel be expanded as an infinite symmetric series in $ Q_{m,n} $, and what is the form of this expansion?
• RQ4 How are the polynomials $ Q_{m,n}(x,y|t,q) $ related to Al-Salam–Chihara polynomials, and what rational function structure do they exhibit in $ t $ and $ q $?
• RQ5 What is the role of $ q $-binomial identities and $ q $-exponential generating functions in deriving symmetric expansions and orthogonal decompositions?

Key findings

• The polynomials $ Q_{m,n}(x,y|t,q) $ are rational functions in $ t $ and $ q $, and can be expressed as linear combinations of Al-Salam–Chihara polynomials, establishing a bridge between univariate and bivariate $ q $-orthogonal systems.
• The spaces $ \Lambda_n(x,y|t,q) = \text{span}\{ Q_{i,n-i}(x,y|t,q) \}_{i=0}^n $ are orthogonal with respect to the $(t,q)$-Normal measure on $ S(q) \times S(q) $, forming a direct sum decomposition of $ L^2 $.
• A symmetric infinite expansion of the reciprocal of the Poisson-Mehler kernel is derived: $ \frac{1}{\gamma_{0,0}(x,y|\rho,q)} = \sum_{n=0}^\infty \frac{\rho^n}{(\rho^2)_\infty} \sum_{k=0}^n (-1)^k q^{\binom{n-k}{2}} \binom{n}{k}_q \frac{\prod_{i=0}^{k-1} \omega(\cdot | \rho q^i)}{(\rho^2)_{2k}} $.
• The generating function for $ Q_{m,n} $ is derived as $ \sum_{i,j \geq 0} \frac{s^i t^j}{[i]_q! [j]_q!} Q_{i,j}(x,y|t,q) = \frac{\varphi_H(x|t,q) \varphi_H(y|s,q)}{\gamma_{0,0}(x,y|t,q)} \sum_{n \geq 0} \frac{t^n}{[n]_q!} H_n(x|t,q) H_n(y|s,q) $, linking to $ q $-exponential generating functions.
• A symmetric identity is proven: $ \sum_{k=0}^n (-1)^k q^{\binom{n-k}{2}} \binom{n}{k}_q \frac{\prod_{i=0}^{k-1} \omega(\cdot | \rho q^i)}{(\rho^2)_{2k}} = q^{\binom{n}{2}} \rho^n (1-q)^n Q_{n,n}(x,y|\rho,q) $, using $ q $-binomial theorems and orthogonality.
• The $(t,q)$-Normal measure is constructed from $ q $-Hermite and Al-Salam–Chihara measures, and its density is explicitly defined on the square $ S(q) \times S(q) $, generalizing the bivariate normal distribution in the $ q $-setting.