On Asymptotics, Stirling Numbers, Gamma Function and Polylogs

Abstract.We apply the Euler–Maclaurin formula to find the asymptotic expansion of the sums $$\sum\nolimits_{k=1}^n (\log k)^p/k^q$$, $$\sum k^q(\log k)^p$$, $$\sum(\log k)^p/(n- k)^q$$, $$\sum 1/k^q(\log k)^p$$ in closed form to arbitrary order $$(p, q \in \mathbb{N})$$. The expressions often simplify considerably and the coefficients are recognizable constants. The constant terms of the asymptotics are either $$\zeta^{(p)}({\pm}q)$$ (first two sums), 0 (third sum) or yield novel mathematical constants (fourth sum). This allows numerical computation of $$\zeta^{(p)}({\pm}q)$$ faster than any current software. One of the constants also appears in the expansion of the function $$\sum\nolimits_{n\geqslant 2}(n \log n)^{-s}$$ 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.