关于渐近性、Stirling数、Γ函数和多对数
On asymptotics, Stirling numbers, Gamma function and polylogs
摘要：
本文应用Euler-Maclaurin公式，求出了和$\sum_{k=1}^n (\log k)^p / k^q$，~ $\sum k^q (\log k)^p$，~ $\sum (\log k)^p /(n-k)^q$，~ $\sum 1/k^q (\log k)^p $的封闭形式到任意阶的渐近展开式（$p,q \in\N$）.这些表达式通常相当简化，并且系数是可识别的常数。渐近的常数项是$ζ^{(p)}(\pm q)$（前两个和）、0（第三个和）或产生新的数学常数（第四个和）。这使得$ζ^{(p)}(\pm q)$的数值计算速度比任何当前软件都快。其中一个常数也出现在函数$\sum_{n\geq 2} (n\log n)^{-s}$在$s=1$处奇点附近的展开式中;这需要不完全Γ函数的渐近性。操作涉及polylogs，我们找到了一个表示的尼尔森积分，以及神秘的伯努利数。应用包括$(-z/\log(1-z))^k$的泰勒系数的渐近增长的测定。给出了第一类Stirling数的渐近性及其用调和数表示的公式。
英文摘要：
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.
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