{"paper":{"title":"Bochner-Pearson-type characterization of the free Meixner class","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OA"],"primary_cat":"math.CO","authors_text":"Michael Anshelevich","submitted_at":"2009-09-06T17:45:45Z","abstract_excerpt":"The operator $L_\\mu: f \\mapsto \\int \\frac{f(x) - f(y)}{x - y} d\\mu(y)$ 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 $p (L_\\nu L_\\mu) - q L_\\mu$ has orthogonal polynomial eigenfunctions is when $\\mu$ and $\\nu$ are related by a Jacobi shift."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.1097","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}