Series involving Fibonacci Numbers: $\sum_{k=1}^\infty \frac{1}{F_kF_{k+1}}$

A note.

The recurrence $\enspace\displaystyle \Phi_n(x)=\frac{\Phi_1(x)}{F_n}-\frac{F_{n-1}}{F_n}\frac{\Phi_{n-1}(x)}{x}+\frac{F_{n-1}x}{F_n^2}\,$ multiplicated with $\,F_n\enspace$ and

$\,n\to\infty\,$ leads to $\,\displaystyle \Phi_1(x)=-\frac{2x}{1+\sqrt{5}}+\sqrt{5}(1+\frac{1}{x})\sum\limits_{k=1}^\infty \frac{x^{k+1}}{(\frac{3+\sqrt{5}}{2})^k-(-1)^k}\enspace$ and therefore

to $\enspace\displaystyle \Phi_1(1)=-\frac{2}{1+\sqrt{5}}+2\sqrt{5}\, f(\frac{3+\sqrt{5}}{2})\enspace$ with $\enspace\displaystyle f(x):=\sum\limits_{k=1}^\infty \frac{1}{x^k-(-1)^k} \,$ , $\enspace |x|>1\,$.

It remains the problem to simplify $f(x)$ which is independend of the Fibonacci numbers.

$\displaystyle f(-\frac{1}{x}) = x\frac{d}{dx}\ln g(x) \enspace$ for $\enspace g(x):=\prod\limits_{k=1}^\infty (1-x^k)^{\frac{(-1)^{k-1}}{k}}\,$ with $\,|x|<1\,$ .

I don't know if $\,g(x)\,$ is easier or worse to discuss than Euler's pentagonal number theorem, based on the Jacobi triple product (e.g. https://en.wikipedia.org/wiki/Jacobi_triple_product).

Another possibility is to discuss $\enspace\displaystyle h(x):=\sum\limits_{k=1}^\infty\frac{x^k}{F_k}\enspace$ for $\,\displaystyle |x|<\frac{1+\sqrt{5}}{2}\enspace$ because of

$\,\displaystyle \Phi_1(x)=-\frac{2x}{1+\sqrt{5}}+(1+x)\,h(\frac{2x}{1+\sqrt{5}})\,$ . $\,$ Sorry, I haven't seen any literature about $\,h(x)\,$ .

About the special case, the irrational reciprocal Fibonacci constant $\,\psi=h(1)\,$ (https://en.wikipedia.org/wiki/Fibonacci_number), is said that there is no closed formula known.

I think we can assume, that there is no closed formula for $\,\Phi_1(x)\,$ . (But maybe it's possible to find an integral for that function which would be interesting.)