{"paper":{"title":"A completely monotonic function involving the tri- and tetra-gamma functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Bai-Ni Guo, Feng Qi","submitted_at":"2010-01-26T08:54:38Z","abstract_excerpt":"The psi function $\\psi(x)$ is defined by $\\psi(x)=\\frac{\\Gamma'(x)}{\\Gamma(x)}$ and $\\psi^{(i)}(x)$ for $i\\in\\mathbb{N}$ denote the polygamma functions, where $\\Gamma(x)$ is the gamma function. In this paper we prove that a function involving the difference between $[\\psi'(x)]^2+\\psi''(x)$ and a proper fraction of $x$ is completely monotonic on $(0,\\infty)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.4611","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"}