A closed form of the family of series $\sum _{k=1}^{\infty } \frac{\left(H_k\right){}^m-(\log (k)+\gamma )^m}{k}$ for $m\ge 1$

2017-12-28 00:46:28

Introduction

Inspired by the work of Olivier Oloa [1] and the question of Vladimir Reshetnikov in a comment I succeeded in calculating the closed form of the sum

$$s(m) = \sum _{k=1}^{\infty } \frac{\left(H_k\right){}^m-(\log (k)+\gamma )^m}{k}\tag{1}$$

for $m=3$.

The cases known (to me) are

$$s(1) = -\gamma _1+\frac{\pi ^2}{12}-\frac{\gamma ^2}{2}= 0.728694...\tag{2a}$$

$$s(2) = -2 \gamma \gamma _1-\gamma _2+\frac{5 \zeta (3)}{3}-\frac{2 \gamma ^3}{3}= 1.96897 ...\tag{2b}$$

$$s(3) =-3 \gamma ^2 \gamma _1-3 \gamma \gamma _2-\gamma _3+\frac{43 \pi ^4}{720}-\frac{3 \gamma ^4}{4}=5.82174 ...\tag{2c}$$

Here $\gamma$ is Euler's gamma, and $\gamma_k$ is Stieltjes gamma of order k.

Notice that $s(2)$ was calculated in [1]. For a check of my method I calculated all three cases.

Inspecting the available cases $m=1, 2, 3$ I tentatively propose here a general formula for the closed form of $s(m)$.

$$ s_{p}(m)= -m \sum _{j=1}^{m-1} \gamma ^j \gamma _{m-j}+a(m) \zeta (m+