Orthogonal polynomials with a resolvent-type generating function

Abstract

Abstract. The subject of this paper are polynomials in multiple non-commuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favard-type recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in non-commuting variables with a resolvent-type generating function. Among such families, we describe the ones that are orthogonal. Their recursion relations have a more special form; the best way to describe them is in terms of the free cumulant generating function of the state of orthogonality, which turns out to satisfy a type of secondorder difference equation. If the difference equation is in fact first order, and the state is tracial, we show that the state is necessarily a rotation of a free product state. We also describe interesting examples of non-tracial infinitely divisible states with orthogonal free Sheffer polynomials.

Meixner Matrix Ensembles

“…The free Meixner family of laws again appeared as the laws characterized by a quadratic regression property in free probability [10], in a class of Markov processes, and as generating measures of Cauchy-Stieltjes kernel families with quadratic variance function [11]. See also [5], [6] and [12,Theorem 4.3]. Except for the free binomial law, the remaining free-Meixner laws are infinitely divisible with respect to free additive convolution and appear as limit laws of large dimensional random matrices [8,13].…”

Free exponential families as kernel families

“…up to "the type") the generating measure ν is: The laws in (i)-(v) are infinitely divisible with respect to free additive convolution (we recall the definition near (3.17)). In [1, Theorem 4] they appear in connection to martingale polynomials with respect to free Lévy processes; free infinite divisibility is analyzed also in [21]; [2] studies further free probability aspects of this family; in [9, Theorem 3.2] the same laws appear as a solution to a quadratic regression problem in free probability; in [11,Theorem 4.3] these laws occur in a "classical regression" problem.…”

Applicability of multiplicative renormalization method for a certain function

“…includes those distributions obtained in [2,5,7,9], in particular, the arcsine and semi-circle distributions. The purpose of the present paper is to solve the above problem for the case when h(x) is given by…”