{"paper":{"title":"Two applications of polylog functions and Euler sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guy Louchard","submitted_at":"2017-09-25T19:22:04Z","abstract_excerpt":"Let $I(n):=\\int_0^1 [x^n+(1-x)^n]^\\frac1n dx.$ In this paper, we show that $I(n)= \\sum_0^\\infty \\frac{I_i}{n^i},n\\rightarrow \\infty$ and we compute $I_i, i =0..5$, obtained by polylog functions and Euler sums. As a corollary, we obtain explicit expressions for some integrals involving functions $ u^i, exp(-u), (1 +exp(-u))^j , ln(1 + exp(-u))^k$ . As another asymptotic result, let $S_0(z):=\\frac{Li_m(1)}{Li_m(1)-Li_m(z)}$, where $Li_m(z)$ is the polylog function. We provide the asymptotic behaviour of $S_n,n\\rightarrow \\infty$ where $S_n:=[z^n]S_0(z)$. This paper fits within the framework of a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.08686","kind":"arxiv","version":2},"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"}