Expansions of one density via polynomials orthogonal with respect to the other

TL;DR

This paper develops expansions of certain densities using orthogonal polynomials, enabling deeper analysis of these functions and revealing new properties, including simplified proofs of classical kernels.

Contribution

It introduces novel expansions of densities in terms of orthogonal polynomial bases, connecting different polynomial families and densities for advanced analysis.

Findings

Expanded densities in terms of orthogonal polynomials.

Provided a simple proof of the Poisson--Mehler kernel expansion.

Discovered new properties of q-Normal and related densities.

Abstract

We expand the Chebyshev polynomials and some of its linear combination in linear combinations of the q-Hermite, the Rogers (q-utraspherical) and the Al-Salam--Chihara polynomials and vice versa. We use these expansions to obtain expansions of some densities, including q-Normal and some related to it, in infinite series constructed of the products of the other density times polynomials orthogonal to it, allowing deeper analysis and discovering new properties. On the way we find an easy proof of expansion of the Poisson--Mehler kernel as well as its reciprocal. We also formulate simple rule relating one set of orthogonal polynomials to the other given the properties of the ratio of the respective densities of measures orthogonalizing these polynomials sets.