Community,
I am new to this forum but found some of your posts very interesting in the past. I'm stuck in a problem related to continued fractions. In particular, the continued fraction is perturbed by the elements of $y_1$.
My problem: Image link
Is there any possible way to derive closed form expressions of $g$ and $h$ in terms of $x$, $y_1$, and $y_2$? There are $n$ fractions.
\begin{equation} \cfrac{g}{h}=\cfrac{\frac{x}{y_1}}{1-\cfrac{x}{y_1}\cfrac{\frac{x}{y_2}}{1-\cfrac{x}{y_2}\cfrac{\frac{x}{y_2}}{\ddots \phantom{=}\cfrac{\vdots}{1-\cfrac{x}{y_2}\cfrac{\frac{x}{y_2}}{1-\cfrac{x}{y_2}\cfrac{\frac{x}{y_2}}{1-\cfrac{x}{y_1}\cfrac{x}{y_2}}}}}}} \end{equation} I would be thankful for any hint or advice. Thank you in advance.
Best regards Josef