On the generalized Kesten–McKay distributions^*

Warsaw University of Technology, Department of Mathematics and Information Sciences, Warszawa, Poland.

^** Corresponding author: pawel.szablowski@gmail.com

Received: 24 March 2016
Accepted: 28 October 2019

Abstract

We examine the properties of distributions with the density of the form: where c, a₁, …, a_n are some parameters and A_n a suitable constant. We find general forms of A_n, 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, …, a_n forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases n = 2, 4, 6.