{"paper":{"title":"The asymptotic expansion of a generalisation of the Euler-Jacobi series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"R.B. Paris","submitted_at":"2015-03-25T10:48:06Z","abstract_excerpt":"We consider the asymptotic expansion of the sum \\[S_p(a;w)=\\sum_{n=1}^\\infty n^{-w}\\e^{-an^p}\\] as $a\\rightarrow 0$ in $|\\arg\\,a|<\\pi/2$ for arbitrary finite $p>$ and $w>0$. Our attention is concentrated mainly on the case when $p$ and $w$ are both even integers, where the expansion consists of a {\\it finite} algebraic expansion together with a sequence of increasingly subdominant exponential expansions. This exponentially small component produces a transformation for $S_p(a;w)$ analogous to the well-known Poisson-Jacobi transformation for the sum with $p=2$ and $w=0$. Numerical results are gi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.07329","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"}