Closed form expressions for harmonic sums I.

Question

This question is a continuation of the topic in Closed form expressions for harmonic sums . By using the integral representation of harmonic numbers and by induction we have derived the following integral representation of an infinite harmonic sum. We have: \begin{equation} {\mathfrak S}^{(p)}_n(t):=\sum\limits_{m=0}^\infty H_m^p \cdot \frac{t^{m+1}}{(m+1)^n} = \int\limits_{[0,1]^p} \prod\limits_{\eta=0}^{p-1} \log\left(1-\xi_\eta\right) \cdot \frac{\left(\sum\limits_{l=0}^{p}(-1)^l \binom{p}{l} Li_{n-l}(t\prod\limits_{\eta=0}^{p-1} \xi_\eta )\right)}{\prod\limits_{\eta=0}^{p-1} \xi_\eta^2} \prod\limits_{\eta=0}^{p-1} d \xi_\eta \end{equation} Here $Li_n()$ are poly-logarithms, $p$ and $n$ are strictly positive integers and $t\in (-1,1)$. Now the question is how can we use the result above to calculate those sums at unity,i.e. how do we calculate ${\mathfrak S}^{(n)}_p(1)$ ? Does the result always depend only on certain values of the zeta function (as it does in case $p=1$ as shown in the question quoted above) or instead does it also depend on something else?