Bochner-Pearson-type characterization of the free Meixner class

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.