6 research outputs found
Cauchy-Stieltjes families with polynomial variance functions and generalized orthogonality
This paper studies variance functions of Cauchy-Stieltjes Kernel families
generated by compactly supported centered probability measures. We describe
several operations that allow us to construct additional variance functions
from known ones. We construct a class of examples which exhausts all cubic
variance functions, and provide examples of polynomial variance functions of
arbitrary degree. We also relate Cauchy-Stieltjes Kernel families with
polynomial variance functions to generalized orthogonality.
Our main results are stated solely in terms of classical probability; some
proofs rely on analytic machinery of free probability.Comment: Minor typos correcte
Densities of the Raney distributions
We prove that if and then the sequence
, , is positive definite, more
precisely, is the moment sequence of a probability measure with
compact support contained in . This family of measures encompasses
the multiplicative free powers of the Marchenko-Pastur distribution as well as
the Wigner's semicircle distribution centered at . We show that if
is a rational number, , then is absolutely continuous and
its density can be expressed in terms of the Meijer and the
generalized hypergeometric functions. In some cases, including the
multiplicative free square and the multiplicative free square root of the
Marchenko-Pastur measure, turns out to be an elementary function
The probability measure corresponding to 2-plane trees
We study the probability measure for which the moment sequence is
. We prove that is absolutely continuous,
find the density function and prove that is infinitely divisible with
respect to the additive free convolution
Orthogonal polynomials induced by discrete-time quantum walks in one dimension
In this paper we obtain some properties of orthogonal polynomials given by a
weight function which is a limit density of a rescaled discrete-time quantum
walk on the line.Comment: 11 pages, this arXiv version has no figure