Closed form for $x+2^ax^2+3^ax^3+...+n^ax^n$

Question

I was wondering if there was a closed form for $$f(x)=x+2^ax^2+3^ax^3+...+n^ax^n+...$$ I have tried to find one but I had no luck. If you divide by $x$ and then integrate you get $$\int\frac{f(x)}{x}dx=x+2^{a-1}x^2+3^{a-1}x^3+...n^{a-1}x^n+...$$ If you repeat this process $a-1$ more times you get the geometric series $$1+x+x^2+...+x^n+...=(1-x)^{-1}$$ but I don't know where to go from here. Thanks.

Answer 1

Evidently, this is called a polylogarithm $$ \sum _{n=1}^\infty n^ax^n=:\mathrm{Li}_{-a}(x). $$

That this has been given a special name implies that there is likely no closed form for it, though the Wikipedia article lists many properties that you may find of use.