  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).
