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

Part of: Basic hypergeometric functions Miscellaneous applications of functional analysis Multivariate analysis Special classes of linear operators

Published online by Cambridge University Press:  02 December 2024

Paweł J. Szabłowski Open the ORCID record for Paweł J. Szabłowski [Opens in a new window]
Paweł J. Szabłowski*
Affiliation:
Faculty of Mathematics and Information Sciences, Warsaw University of Technology, ul Koszykowa 75, 00-662 Warsaw, mazowieckie, Poland ([email protected]) (corresponding author)
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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 (LEs) of the bivariate distributions absolutely continuous with respect to the product of their marginal distributions. Applications of LE 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).

Keywords

q-Hermite polynomialsq-seriesq-ultrasphericalbivariate distributionHermiteLancaster expansionsorthogonal polynomials

MSC classification

Primary: 62H05: Characterization and structure theory 33D45: Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
Secondary: 47B34: Kernel operators 46N30: Applications in probability theory and statistics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 34
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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