On Asymptotics, Stirling Numbers, Gamma Function and Polylogs

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.

Authors and Affiliations

1. MPI Bonn, Vivatsgasse 7, D-53111, Bonn, Germany

   Daniel B. Grünberg

Corresponding author

Correspondence to Daniel B. Grünberg.

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