On the generalized Kesten--McKay distributions

Abstract

We examine the properties of distributions with the density of the form: where $c,a_1,\dots ,a_n$ are some parameters and $A_n$ a suitable constant. We find general forms of , 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 $a_1,\dots ,a_n$ forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases $n=2,4,6.$

14 pages

References in corpus (6)

• Expansions of one density via polynomials orthogonal with respect to the other
• On the structure and probabilistic interpretation of Askey-Wilson densities and polynomials with complex parameters
• Moments of q-Normal and conditional q-Normal distributions
• On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures
• Befriending Askey-Wilson polynomials
• Around Poisson--Mehler summation formula