On asymptotics, Stirling numbers, Gamma function and polylogs
Abstract
We apply the Euler--Maclaurin formula to find the asymptotic expansion of the sums
∑
n
k=1
(logk
)
p
/
k
q
, ~
∑
k
q
(logk
)
p
, ~
∑(logk
)
p
/(n−k
)
q
, ~
∑1/
k
q
(logk
)
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)
(±q)
(first two sums), 0 (third sum) or yield novel mathematical constants (fourth sum). This allows numerical computation of
ζ
(p)
(±q)
faster than any current software. One of the constants also appears in the expansion of the function
∑
n≥2
(nlogn
)
−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.