# A family of generating functions related to generalized harmonic numbers and power functions.

We want to generalize the problem Strategies for evaluating sums $\sum_{n=1}^\infty \frac{H_n^{(m)}z^n}{n}$ . To be precise we want to find closed form expressions for sums of the kind: $${\mathfrak S}_q^{(m)}(x) := \sum\limits_{k=1}^\infty \frac{H^{(m)}_k}{k^q} x^k$$ where $H^{(m)}_k$ are generalized harmonic numbers and $\xi \in [-1,1)$. Now, by using the generating function of generalized harmonic numbers we have shown the following: $${\mathfrak S}_q^{(1)}(x) = Li_{q+1}(x) + S_{q-1,2}(x)$$ for $q=1,2,\cdots$. Here $Li_q(x)$ are polylogarithms and $S_{q,p}(x)$ are Nielsen generalized polylogarithms. Unfortunately for $m=2$ the situation becomes more complicated. We have: \begin{eqnarray} {\mathfrak S}^{(2)}_1(x) &=& Li_3(x) - \log(1-x) Li_2(x) - 2 S_{1,2}(x) \\ {\mathfrak S}^{(2)}_2(x) &=& Li_4(x) + \frac{1}{2} Li_2(x)^2 - 2 S_{2,2}(x) \\ {\mathfrak S}^{(2)}_3(x) &=& Li_5(x) + \frac{1}{2} \int\limits_0^x \frac{[Li_2(\xi)]^2}{\xi} d\xi - 2 S_{3,2}(x) \\ \vdots \end{eqnarray} Now we got stuck. At $m=2$ and $q=3$ we come across a term(the integral on the right hand side above) that we have been unable to express in terms of polylogarithms, Nielsen polylogarithms or elementary functions. The question is therefore is it possible to express that term in terms of the functions in question and if it is not possible can we prove that impossibility.

Note: The definite integral in question has been computed in here Definite Dilogarithm integral $\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx$ .

Note1: A similar question has been asked in here Tough quadrilogarithm integral but no satisfactory answer was found.