Regarding question number 2, you can write $f_k = \frac{1}{\sqrt{5}} \left( \phi^k - \left(1-\phi\right)^k \right)$, where $\phi^2 = \phi + 1$. Hence
$$\sum_{k=1}^{\infty} \frac{f_k}{k^s} = \frac{1}{\sqrt5} \sum_{k=1}^{\infty} \frac{\phi^k - \left(1-\phi\right)^k}{k^s}$$
By the definition of the [Polylogarithm](https://en.wikipedia.org/wiki/Polylogarithm), we get
$$\sum_{k=1}^{\infty} \frac{f_k}{k^s} = \frac{1}{\sqrt5}\mathrm{Li}_s\left( \phi \right) - \frac{1}{\sqrt5}\mathrm{Li}_s\left( 1-\phi \right)$$