{"paper":{"title":"On the asymptotic expansions of products related to the Wallis, Weierstrass and Wilf formulas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"C.-P. Chen, R.B. Paris","submitted_at":"2015-11-30T09:40:23Z","abstract_excerpt":"For all integers $n\\geq1$, let \\begin{align*}\n  W_n(p,q)=\\prod_{j=1}^{n}\\left\\{e^{-p/j}\\left(1+\\frac{p}{j}+\\frac{q}{j^2}\\right)\\right\\} \\end{align*} and \\begin{align*} R_n(p, q)=\\prod_{j=1}^{n}\\left\\{e^{-p/(2j-1)}\\left(1+\\frac{p}{2j-1}+\\frac{q}{(2j-1)^2}\\right)\\right\\}, \\end{align*} where $p$, $q$ are complex parameters. The infinite product $W_{\\infty}(p,q)$ includes the Wallis and Wilf formulas, and also the infinite product definition of Weierstrass for the gamma function, as special cases. In this paper, we present asymptotic expansions of $W_n(p,q)$ and $R_n(p, q)$ as $n\\to\\infty$. In add"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.09217","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"}