I'm given a generating function $$G(x) = \sum\limits_{k=0}^{\infty}a_k x^k$$ for a sequence $(a_0, a_1, a_2, \ldots)$.
I know, that I can express generating functions for sequences like $(a_0 + c, a_1 + c, a_2 + c, \ldots)$ or $(1 \cdot a_1, 2 \cdot a_2, 3 \cdot a_3, \ldots)$ in terms of $G(x)$. For example $$xG^{'}(x) = \sum\limits_{k=0}^{\infty} k a_k x^k$$
So I'm interested,
could something be done to express the function $$F(x) = \sum\limits_{k=0}^{\infty} \sqrt{k} a_k x^k$$ in terms of $G(x)$ and/or its derivatives?
Consider the function $$ G(x) = \sum_{n=1}^\infty x^n = \frac{x}{1-x} $$ This, and all its derivatives, are rational functions. But (as vrungtehagel noted) $$ F(x) = \sum_{n=1}^\infty \sqrt{n}\;x^n = \mathrm{Li}_{-1/2}(x) , $$ is a polylogarithm (of non-integer type), so certainly not a rational function.
You may think of the problem as trying to do the differential operator $$ x\;\frac{d}{dx} $$ a fractional numer of times: $$ F(x) = \left(x\frac{d}{dx}\right)^{1/2} G(x) \tag{?} $$