Bochner-Pearson-type characterization of the free Meixner class

The operator Lμ:f↦∫f(x)−f(y)x−ydμ(y) is, for a compactly supported measure μ with an L3 density, a closed, densely defined operator on L2(μ). We show that the operator Q=pLμ2−qLμ has polynomial eigenfunctions if and only if μ is a free Meixner distribution. The only time Q has orthogonal polynomial eigenfunctions is if μ is a semicircular distribution. More generally, the only time the operator p(LνLμ)−qLμ has orthogonal polynomial eigenfunctions is when measures μ and ν are related by a Jacobi shift.