Two applications of polylog functions and Euler sums

2017/09/25 by Louchard, Guy
#05A16 #60C05 #60F05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1709.08686

Abstract

Let I(n):=∫₀¹ [xⁿ+(1-x)ⁿ]^\frac1n dx. In this paper, we show that I(n)= ∑₀^∞ (I_i)/(nⁱ),n→ ∞ 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 uⁱ, exp(-u), (1 +exp(-u))^j , ln(1 + exp(-u))^k . As another asymptotic result, let S₀(z):=(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→ ∞ where S_n:=[zⁿ]S₀(z). This paper fits within the framework of analytic combinatorics.

Related

• On asymptotics, Stirling numbers, Gamma function and polylogs
• On Louchard's Asymptotic Series
• An Asymptotic Series for an Integral
• Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums
• Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series