Volume 24, 2020
|Page(s)||56 - 68|
|Published online||27 February 2020|
On the generalized Kesten–McKay distributions*
Warsaw University of Technology, Department of Mathematics and Information Sciences,
** Corresponding author: email@example.com
Accepted: 28 October 2019
We examine the properties of distributions with the density of the form: where c, a1, …, an are some parameters and An a suitable constant. We find general forms of An, of k-th moment and of k-th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so called Askey–Wilson scheme. On the way we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters a1, …, an forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases n = 2, 4, 6.
Mathematics Subject Classification: 60E05 / 05E05 / 62H05 / 60J10
Key words: Kesten–McKay / Bernstein-Szegö distributions / Chebyshev polynomials / orthogonal polynomials / Askey–Wilson polynomials / moments / symmetric rational functions / multivariate distributions / Cauchy (Hilbert) transform
© EDP Sciences, SMAI 2020
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