{"paper":{"title":"Real-Root Preserving Differential Operator Representations of Orthogonal Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"David A. Cardon, Evan L. Sorensen, Jason C. White","submitted_at":"2017-07-17T23:27:08Z","abstract_excerpt":"In this paper, we study linear transformations of the form $T[x^n]=P_n(x)$ where $\\{P_n(x)\\}$ is an orthogonal polynomial system. Of particular interest is understanding when these operators preserve real-rootedness in polynomials. It is known that when the $P_n(x)$ are the Hermite polynomials or standard Laguerre polynomials, the transformation $T$ has this property. It is also known that the transformation $T[x^n]=H_n^{\\alpha}(x)$, where $H_n^{\\alpha}(x)$ is the $n$th generalized Hermite Polynomial with real parameter $\\alpha$, has the differential operator representation $T[x^n]=e^{-\\frac{\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.05412","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"}