On the generalized Kesten--McKay distributions

TL;DR

This paper explores a class of generalized Kesten--McKay distributions, deriving their properties, moments, orthogonal polynomials, and connections to the Askey--Wilson scheme, with implications for multivariate interpretations.

Contribution

It introduces a comprehensive analysis of these distributions, including explicit formulas and connections to known polynomial schemes, expanding understanding of their mathematical structure.

Findings

Derived general forms of normalization constants and moments.

Calculated Cauchy transforms for the distributions.

Established links to Askey--Wilson polynomials and multivariate cases.

Abstract

We examine the properties of distributions with the density of the form: $% \frac{2A_{n}c^{n-2}\sqrt{c^{2}-x^{2}}}{\pi \prod_{j=1}^{n}(c(1+a_{j}^{2})-2a_{j}x)},$ where $c,a_{1},\ldots ,a_{n}$ 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 $a_{1},\ldots ,a_{n}$ forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases $n=2,4,6.$