Bochner-Pearson-type characterization of the free Meixner class
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abstract
The operator $L_mu: f mapsto int frac{f(x) - f(y)}{x - y} dmu(y)$ 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 $p (L_ u L_mu) - q L_mu$ has orthogonal polynomial eigenfunctions is when $mu$ and $ u$ are related by a Jacobi shift.
