On asymptotics, Stirling numbers, Gamma function and polylogs

Abstract

We apply the Euler--Maclaurin formula to find the asymptotic expansion of the sums $\sum_{k=1}^n (\log k)^p / k^q$, ~$\sum k^q (\log k)^p$, ~$\sum (\log k)^p /(n-k)^q$, ~$\sum 1/k^q (\log k)^p $ in closed form to arbitrary order ($p,q \in\N$). The expressions often simplify considerably and the coefficients are recognizable constants. The constant terms of the asymptotics are either $ζ^{(p)}(\pm q)$ (first two sums), 0 (third sum) or yield novel mathematical constants (fourth sum). This allows numerical computation of $ζ^{(p)}(\pm q)$ faster than any current software. One of the constants also appears in the expansion of the function $\sum_{n\geq 2} (n\log n)^{-s}$ around the singularity at $s=1$; this requires the asymptotics of the incomplete gamma function. The manipulations involve polylogs for which we find a representation in terms of Nielsen integrals, as well as mysterious conjectures for Bernoulli numbers. Applications include the determination of the asymptotic growth of the Taylor coefficients of $(-z/\log(1-z))^k$. We also give the asymptotics of Stirling numbers of first kind and their formula in terms of harmonic numbers.