{"paper":{"title":"Unique positive solution for an alternative discrete Painlev\\'e I equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"nlin.SI","authors_text":"Ana F. Loureiro, Peter A. Clarkson, Walter Van Assche","submitted_at":"2015-08-20T08:19:18Z","abstract_excerpt":"We show that the alternative discrete Painlev\\'e I equation (alt-dP$_{\\rm I}$) has a unique solution which remains positive for all $n \\geq 0$. Furthermore, we identify this positive solution in terms of a special solution of the second Painlev\\'e equation (P$_{\\rm II}$) involving the Airy function $\\mathop{\\rm Ai}(t)$. The special-function solutions of P$_{\\rm II}$ involving only the Airy function $\\mathop{\\rm Ai}(t)$ therefore have the property that they remain positive for all $n\\geq 0$ and all $t \\geq 0$, which is a new characterization of these special solutions of P$_{\\rm II}$ and alt-dP"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.04916","kind":"arxiv","version":2},"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"}