Bochner–Pearson-type characterization of the free Meixner class Academic Article uri icon


  • The operator Lμf(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-q Lμ 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. © 2010 Elsevier Inc. All rights reserved.

author list (cited authors)

  • Anshelevich, M.

citation count

  • 6

publication date

  • January 2011