On the generalized Kesten–McKay distributions

Abstract

We examine the properties of distributions with the density of the form: [see formula in PDF] 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.