{"paper":{"title":"Linear Difference Equations with a Transition Point at the Origin","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Lihua Cao, Yutian Li","submitted_at":"2013-03-20T06:24:46Z","abstract_excerpt":"A pair of linearly independent asymptotic solutions are constructed for the second-order linear difference equation {equation*} P_{n+1}(x)-(A_{n}x+B_{n})P_{n}(x)+P_{n-1}(x)=0, {equation*} where $A_n$ and $B_n$ have asymptotic expansions of the form {equation*} A_n\\sim n^{-\\theta}\\sum_{s=0}^\\infty\\frac{\\alpha_s}{n^s},\\qquad B_n\\sim\\sum_{s=0}^\\infty\\frac{\\beta_s}{n^s}, {equation*} with $\\theta\\neq0$ and $\\alpha_0\\neq0$ being real numbers, and $\\beta_0=\\pm2$. Our result hold uniformly for the scaled variable $t$ in an infinite interval containing the transition point $t_1=0$, where $t=(n+\\tau_0)^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.4846","kind":"arxiv","version":3},"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"}