Asymptotic expansions of integrals and Nielsen's polylogarithms

TL;DR

This paper develops full asymptotic expansions for a class of integrals involving parameters and relates the coefficients to Nielsen's polylogarithms, revealing connections to multiple zeta values and symmetry properties.

Contribution

It derives explicit asymptotic expansions for integrals with parameters, linking coefficients to Nielsen's polylogarithms and multiple zeta values, and explores symmetry conditions affecting these coefficients.

Findings

Coefficients relate to Nielsen's polylogarithms.

For q=-1, expansions involve multiple zeta values.

Symmetry constraints reduce coefficients to polynomials in zeta values.

Abstract

This article derives full asymptotic expansions for integrals of the form \[ \int_{0}^{1}f(u)(1+q\cdot u^{n})^{w/n}du \] as $n\rightarrow\infty$, with parameters real $w\neq 0$ and $q\in(-1,1]$, or positive $w$ for $q=-1$. We relate the coefficients of the asymptotic expansions to Nielsen's generalized polylogarithms. For $q=-1$, we obtain an expansion in terms of multiple zeta values, which in this setting, reduce to ordinary zeta values. A key point is that for $q=1$, the integrals typically produce alternating multiple zeta values; we formulate a precise symmetry constraint on the relevant coefficient sequence under which all coefficients reduce to polynomials in ordinary zeta values. We also translate this symmetry into a statement about a binomial transform, and we verify the condition for several classical Appell-type families, like Euler, Bernoulli, Genocchi, and Hermite. Finally, we obtain precise results about the convergence of norms of random variables.