{"paper":{"title":"Some results associated with Bernoulli and Euler numbers with applications","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":"2016-01-10T09:37:28Z","abstract_excerpt":"In this paper, we present series representations of the remainders in the expansions for $2/(e^t+1)$, $\\mbox{sech} t$ and $\\coth t$.\n  For example, we prove that for $t > 0$ and $N\\in\\mathbb{N}:=\\{1, 2, \\ldots\\}$, \\[\\mbox{sech}\\, t=\\sum_{j=0}^{N-1}\\frac{E_{2j}}{(2j)!}t^{2j}+R_N(t) \\] with \\[ R_N(t)=\\frac{(-1)^{N}2t^{2N}}{\\pi^{2N-1}}\\sum_{k=0}^{\\infty}\\frac{(-1)^{k}}{(k+\\frac{1}{2})^{2N-1}\\Big(t^2+\\pi^2(k+\\frac{1}{2})^2\\Big)}, \\] and \\[\\mbox{sech}\\, t=\\sum_{j=0}^{N-1}\\frac{E_{2j}}{(2j)!}t^{2j}+\\Theta(t, N)\\frac{E_{2N}}{(2N)!}t^{2N} \\] with a suitable $0 < \\Theta(t, N) < 1$. Here $E_n$ are the E"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.02192","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"}