On summable, positive Poisson-Mehler kernels built of Al-Salam--Chihara and related polynomials

TL;DR

This paper derives simple, closed-form expressions for positive, summable kernels constructed from Al-Salam-Chihara and related polynomials, using an expansion technique of density ratios, and explores their properties and special cases.

Contribution

It introduces a novel expansion technique for density ratios to obtain explicit kernels built from ASC and related polynomials, including symmetric and asymmetric positive kernels.

Findings

Derived closed-form kernels from ASC polynomials

Expanded reciprocals of kernels to find polynomial identities

Identified special cases linking to Hermite and Chebyshev polynomials

Abstract

Using special technique of expanding ratio of densities in an infinite series of polynomials orthogonal with respect to one of the densities, we obtain simple, closed forms of certain kernels built of the so called Al-Salam-Chihara (ASC) polynomials. We consider also kernels built of some other families of polynomials such as the so called big continuous q-Hermite polynomials that are related to the ASC polynomials. The constructed kernels are symmetric and asymmetric. Being the ratios of the densities they are automatically positive. We expand also reciprocals of some of the kernels, getting nice identities built of the ASC polynomials involving 6 variables like e.g. formula (nice). These expansions lead to asymmetric, positive and summable kernels. The particular cases (referring to q=1 and q=0) lead to the kernels build of certain linear combinations of the ordinary Hermite and Chebyshev polynomials.