Bochner-pearson-type Characterization of the Free Meixner Class
نویسندگان
چکیده
The operator Lμ : f 7→ ∫ f(x)−f(y) x−y dμ(y) is, for a compactly supported measure μ with an L density, a closed, densely defined operator on L(μ). We show that the operator Q = pLμ−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 μ and ν are related by a Jacobi shift.
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