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Regarding question number 2, you can write $f_k = \frac{1}{\sqrt{5}} \left( \phi^k - \left(1-\phi\right)^k \right)$, where $\phi^2 = \phi + 1$, $\phi > 0$. 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, we get $$\sum_{k=1}^{\infty} \frac{f_k}{k^s} = \frac{1}{\sqrt5}\mathrm{Li}_s\left( \phi \right) - \frac{1}{\sqrt5}\mathrm{Li}_s\left( 1-\phi \right)$$