I would like to take powers of arbitrary order to polylogarithm functions. For instance, given $$ \text{Li}_\alpha(z) = \sum_{k=1}^\infty \frac{z^k}{k^\alpha} $$ I am interested in $$ [\text{Li}_\alpha(z)]^m = \left(\sum_{k=1}^\infty \frac{z^k}{k^\alpha}\right) ^m $$
Of particular interest is how the integral representation
$$ \text{Li}_\alpha(z) = \frac{z}{\Gamma(\alpha)}\int^\infty_0dt \frac{t^{\alpha-1}}{e^t-z} $$ would behave in this case. Does anyone know of research in this area ?
well: $$\ln(1+z)=\sum_{k=0}^\infty\frac{(-1)^{k+1}}{k}x^k$$ so: $$\ln(1-z)=-\sum_{k=0}^\infty\frac{x^k}{k}$$ $$\therefore \operatorname{Li}_1^m(z)=(-1)^m\ln^m(1-z)$$
If we first try and start with a function of the form: $$\frac{d}{dx}\operatorname{Li}_a(x)=\sum_{k=0}^\infty x^{k^a-1}=\frac1x\sum_{k=0}^\infty x^{k^a}$$ maybe there is a way to find a function that is represent by this? :)