Asymptotic expansions of integrals and Nielsen's polylogarithms

This article derives full asymptotic expansions for integrals of the form ∫01f(u)(1+q⋅un)w/n duas n→∞, with parameters real w≠0 and q∈(−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.