Bochner-Pearson-type characterization of the free Meixner class

The operator $L_μ: f \mapsto \int \frac{f(x) - f(y)}{x - y} dμ(y)$ is, for a compactly supported measure $μ$ with an $L^3$ density, a closed, densely defined operator on $L^2(μ)$. We show that the operator $Q = p L_μ^2 - q L_μ$ 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 $p (L_νL_μ) - q L_μ$ has orthogonal polynomial eigenfunctions is when $μ$ and $ν$ are related by a Jacobi shift.

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