Two applications of polylog functions and Euler sums

TL;DR

This paper explores the asymptotic behavior of specific integrals and series involving polylogarithms and Euler sums, providing explicit formulas and asymptotic expansions within analytic combinatorics.

Contribution

It introduces new asymptotic expansions for integrals and series involving polylog functions and Euler sums, with explicit computations for initial terms.

Findings

Derived asymptotic expansion of I(n) for large n

Computed explicit expressions for integrals involving exponential and logarithmic functions

Analyzed the asymptotic behavior of coefficients in polylogarithm series

Abstract

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 analytic combinatorics.