Two applications of polylog functions and Euler sums

Abstract

Let $I(n):={\int }_0^1[x^n+(1-x{)}^n{]}^{\frac{1}{n}}dx.$ In this paper, we show that $I(n)={\sum }_0^{\infty }\frac{I_i}{n^i},n\to \infty$ and we compute $I_i,i=\mathrm{0..5}$, 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 $S_0(z):=\frac{L{i}_m(1)}{L{i}_m(1)-L{i}_m(z)}$, where $L{i}_m(z)$ is the polylog function. We provide the asymptotic behaviour of $S_n,n\to \infty$ where $S_n:=[z^n]S_0(z)$. This paper fits within the framework of analytic combinatorics.

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