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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)^"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1303.4846","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2013-03-20T06:24:46Z","cross_cats_sorted":[],"title_canon_sha256":"a7c4473c10b81e2003f980dc94b7d12b5d6dfd8ff32e921d2de4d59653595c21","abstract_canon_sha256":"757baebd2bf1aac67024bc77e9a3ee848ccf9147ee136572834054bf171d8c64"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:54:39.667774Z","signature_b64":"EbnVNB3KvMiEcKYKuNFN58lMEGrMqM3gf4m7HBp6QC5QMacV7n7QQtW2mRp4GCqpadPrUG3obfgMNABipkooCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ffab793cd11b73213b4cbded7b83512618bc0e59f841d7750e4087ecb1b7a51","last_reissued_at":"2026-05-18T02:54:39.667024Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:54:39.667024Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\beta_0=\\pm2$. 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