Bochner-Pearson-type characterization of the free Meixner class

TL;DR

This paper characterizes the free Meixner class of distributions through an operator involving free probability, showing that polynomial eigenfunctions occur precisely for these distributions, with special cases like the semicircular distribution.

Contribution

It provides a new operator-based characterization of the free Meixner distributions and identifies conditions under which polynomial eigenfunctions and orthogonal polynomial eigenfunctions occur.

Findings

Operator $Q$ has polynomial eigenfunctions iff $ ext{ is a free Meixner distribution}$.

Only the semicircular distribution yields orthogonal polynomial eigenfunctions.

Orthogonal polynomial eigenfunctions occur when $$ and $$ are related by a Jacobi shift.

Abstract

The operator $L_\mu: f \mapsto \int \frac{f(x) - f(y)}{x - y} d\mu(y)$ is, for a compactly supported measure $\mu$ with an $L^3$ density, a closed, densely defined operator on $L^2(\mu)$. We show that the operator $Q = p L_\mu^2 - q L_\mu$ has polynomial eigenfunctions if and only if $\mu$ is a free Meixner distribution. The only time $Q$ has orthogonal polynomial eigenfunctions is if $\mu$ is a semicircular distribution. More generally, the only time the operator $p (L_\nu L_\mu) - q L_\mu$ has orthogonal polynomial eigenfunctions is when $\mu$ and $\nu$ are related by a Jacobi shift.