On a class of bivariate distributions built of q-ultraspherical polynomials

TL;DR

This paper investigates a family of bivariate distributions constructed from q-ultraspherical polynomials, focusing on their positivity, polynomial conditional moments, and potential applications in distribution theory and stochastic processes.

Contribution

It introduces a new class of bivariate distributions with positive kernels and polynomial conditional moments based on q-ultraspherical polynomials, expanding the theory of Lancaster expansions.

Findings

Positivity of kernels constructed from q-ultraspherical polynomials.

The family has polynomial conditional moments.

Connections to Lancaster expansions and applications in stochastic processes.

Abstract

Our primary result concerns the positivity of specific kernels constructed using the $q$-ultraspherical polynomials. In other words, it concerns a two-parameter family of bivariate, compactly supported distributions. Moreover, this family has a property that all its conditional moments are polynomials in the conditioning random variable. The significance of this result is evident for individuals working on distribution theory, orthogonal polynomials, $q$-series theory, and the so-called quantum polynomials. Therefore, it may have a limited number of interested researchers. That is why, we put our results into a broader context. We recall the theory of Hilbert-Schmidt operators and the idea of Lancaster expansions of the bivariate distributions absolutely continuous with respect to the product of their marginal distributions. Applications of Lancaster expansion can be found in Mathematical Statistics or the creation of Markov processes with polynomial conditional moments (the most well-known of these processes is the famous Wiener process).