BochnerPearson-type characterization of the free Meixner class
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The operator Lf(x)-f(y)x-yd(y) is, for a compactly supported measure with an L3 density, a closed, densely defined operator on L2(). We show that the operator Q=pL2-q L has polynomial eigenfunctions if and only if is a free Meixner distribution. The only time Q has orthogonal polynomial eigenfunctions is if is a semicircular distribution. More generally, the only time the operator p(LL)-qL has orthogonal polynomial eigenfunctions is when measures and are related by a Jacobi shift. 2010 Elsevier Inc. All rights reserved.
Advances in Applied Mathematics
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