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 $Q = p L_\mu^2 - q L_\mu$haspolynomialeigenfunctionsifandonlyif$\mu$ 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.