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

Paweł J. Szabłowski

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).

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

Abstract

Our primary result concerns the positivity of specific kernels constructed using the -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, -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).
Paper Structure (15 sections, 13 theorems, 144 equations)

This paper contains 15 sections, 13 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Notation, definition and some methods of obtaining LE
  3. Method of expansion of the ratio of densities
  4. Families of polynomials appearing in the paper including those forming part of the Askey-Wilson scheme
  5. Chebyshev polynomials
  6. Hermite polynomials
  7. $q$-Hermite polynomials
  8. Big $q$-Hermite polynomials
  9. Continuous $q$-ultraspherical polynomials
  10. Al-Salam--Chihara polynomials.
  11. Positive, summable kernel built of $q$-ultraspherical polynomials
  12. List of Lancaster kernels
  13. Symmetric kernels
  14. Non-symmetric kernels
  15. Complicated proofs and auxiliary facts from $q$-series theory.

Key Result

Lemma 1

1) and where density $f_{R}$ is given by (fR) and moreover, can be presented in one of the following equivalent forms: We also have the following expansion of $f_{N}/f_{R}$ in orthogonal series in polynomials $\left\{ R_{n}\right\} :$ 2) We also have the following linearization formulae

Theorems & Definitions (34)

  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 24 more