{"paper":{"title":"A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.AP"],"primary_cat":"math.CA","authors_text":"Carlos E. Arreche","submitted_at":"2013-04-06T17:40:15Z","abstract_excerpt":"We give simple necessary and sufficient conditions for the $\\frac{\\partial}{\\partial t}$-transcendence of the solutions to a parameterized second order linear differential equation of the form \\frac{\\partial^2 Y}{\\partial x^2} - p \\frac{\\partial Y}{\\partial x} = 0, where $p\\in F(x)$ is a rational function in $x$ with coefficients in a $\\frac{\\partial}{\\partial t}$-field $F$. This result is crucial for the development of an efficient algorithm to compute the parameterized Picard-Vessiot group of an arbitrary parameterized second-order linear differential equation over $F(x)$. Our criteria imply"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.1917","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"}