On the generalized Kesten--McKay distributions

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 $n=2,4,6.$