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Our most important result concerns the positivity of certain kernels built ofthe so-called $q-$ultraspherical polynomials. Since this result appears atfirst sight as primarily important for those who are working in orthogonalpolynomials, $q-$series theory and the so-called quantum polynomials, it mighthave a limited number of interested researchers. That is why, we put our resultinto a broader context. We recall the theory of Hilbert-Schmidt operators,Lancaster expansions and their applications in Mathematical statistics, orbivariate distributions absolutely continuous with respect to the product oftheir marginal distributions leading to the generation of Markov process withpolynomial conditional moments (the main representative of such processes is afamous Wiener process).

On positivity of the two-parameter bivariate kernel built of  q-ultraspherical polynomials and other Lancaster-type expansions of bivariate  distributions