{"paper":{"title":"Analytic continuation of solutions of the pantograph equation by means of $\\theta$-modular formula","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Changgui Zhang","submitted_at":"2012-02-02T12:46:13Z","abstract_excerpt":"The aim of this paper is to treat the constant coefficients functional-differential equation $y'(x)=ay(qx)+by(x)$ with the help of the analytic theory of linear $q$-difference equations. When $ab\\not=0$, the associated Cauchy problem with $y(0)=1$ admits a unique power series solution, which is the Hadamard product of a usual-hypergeometric series by a basic-hypergeometric series. By means of $\\theta$-modular relation, it is proved that this entire function can be expressed as linear combination of all the elements of a system of canonical fundamental solutions at infinity. A family of power s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0423","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"}