{"paper":{"title":"On Hyperbolic Polynomials and Four-term Recurrence with Linear Coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.CV","authors_text":"Richard Adams","submitted_at":"2019-04-29T05:50:47Z","abstract_excerpt":"For any real numbers $a,\\ b$, and $c$, we form the sequence of polynomials $\\{P_n(z)\\}_{n=0}^\\infty$ satisfying the four-term recurrence \\[ P_n(z)+azP_{n-1}(z)+bP_{n-2}(z)+czP_{n-3}(z)=0,\\ n\\in\\mathbb{N}, \\] with the initial conditions $P_0(z)=1$ and $P_{-n}(z)=0$. We find necessary and sufficient conditions on $a,\\ b$, and $c$ under which the zeros of $P_n(z)$ are real for all $n$, and provide an explicit real interval on which $\\displaystyle\\bigcup_{n=0}^\\infty\\mathcal{Z}(P_n)$ is dense, where $\\mathcal{Z}(P_n)$ is the set of zeros of $P_n(z)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.12455","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"}