Asymptotic expansions of integrals and Nielsen's polylogarithms

This article derives full asymptotic expansions for integrals of the form $${\int }_0^{1}f(u)(1+q\cdot u^n{)}^{w\mathrm{/}n}du$$ as $n\to \mathrm{\infty }$, with parameters real $w\mathrm{\ne }0$ 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.