This question is a generalization of How can I prove this closed form for $\sum_{n=1}^\infty\frac{(4n)!}{\Gamma\left(\frac23+n\right)\,\Gamma\left(\frac43+n\right)\,n!^2\,(-256)^n}$ .
Now, let $m\ge 2$ and $k \ge 0$ be integers. Let also $0 < w < (m-1)^{m-1}/m^m$ be a real number. We consider a following family of sums: \begin{equation} {\mathfrak S}^{(m)}_k(w):= \sum\limits_{i=1}^\infty \binom{m\cdot i}{i} \cdot \frac{w^i}{i^k} \end{equation}
Now, by using my answer to Closed form solutions for a family of hypergeometric sums. and then by carrying out the usual operations, i.e. dividing both sides of the equation by $w$ and integrating and repeating this operation several times we have derived the following results: \begin{eqnarray} {\mathfrak S}^{(m)}_0(w) &=& \frac{m w x^{m-1}-w x^m+x-1}{1-m w x^{m-1}}\\ {\mathfrak S}^{(m)}_1(w) &=& m \log(x)\\ {\mathfrak S}^{(m)}_2(w) &=& -\frac{1}{6} m \left(3 m \log ^2(x)-6 \text{Li}_2(x)-6 \log (1-x) \log (x)+\pi ^2\right)\\ {\mathfrak S}^{(m)}_3(w) &=& \frac{1}{6} m \left(m^2 \log ^3(x)-6 m \text{Li}_3(x)-3 m \log (1-x) \log ^2(x)+\pi ^2 m \log (x)+6 m \zeta (3)-6 \text{Li}_3(1-x)\right)\\ {\mathfrak S}^{(m)}_4(w) &=& -\frac{1}{360} m \left(\right.\\ &&\left.+15 (m+1)_{(3)} \log ^4(x)\right.\\ &&\left.-360 (m-1) \text{Li}_4(1-x)-360 m \text{Li}_4\left(\frac{x-1}{x}\right)-360 (m-1) m \text{Li}_4(x)\right.\\ &&\left.+360 (m-1) m \zeta (3) \log (x)\right.\\ &&\left.-60 (m-1) m \log (1-x) \log ^3(x)+30 \pi ^2 (m-1) m \log ^2(x)+4 \pi ^4 (m-1) m\right.\\ &&\left.\right) \end{eqnarray} where $x=x(w)$ is calculated in the following way. Out of the solutions of the equation \begin{equation} 1-x+w \cdot x^m=0 \end{equation} we choose the one that is the closest to unity.
Now of course the question would be to come up with a closed form solution for generic values of $k > 3$. Another question is again, are the poly-logarithms along with elementary functions sufficient to express the result or instead we need some more general family of functions for that purpose, for generic values of $k > 3$?
