{"paper":{"title":"Global Pad\\'e approximations of the generalized Mittag-Leffler function and its inverse","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Caibin Zeng, YangQuan Chen","submitted_at":"2013-10-14T03:01:59Z","abstract_excerpt":"This paper proposes a global Pad\\'{e} approximation of the generalized Mittag-Leffler function $E_{\\alpha,\\beta}(-x)$ with $x\\in[0,+\\infty)$. This uniform approximation can account for both the Taylor series for small arguments and asymptotic series for large arguments. Based on the complete monotonicity of the function $E_{\\alpha,\\beta}(-x)$, we work out the global Pad\\'{e} approximation [1/2] for the particular cases $\\{0<\\alpha<1, \\beta>\\alpha\\}$, $\\{0<\\alpha=\\beta<1\\}$, and $\\{\\alpha=1, \\beta>1\\}$, respectively. Moreover, these approximations are inverted to yield a global Pad\\'{e} approxi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.5592","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"}