I'm using an article that uses Continued fractions to reason an approximation scheme, but I did not found articles or something in literature that could help me reason the steps:
We want to find $\lambda$, we have the recurrence relation: $\alpha_kb_{k+1}+(\tilde{\beta_k}-\lambda)b_{k}+\gamma_kb_{k-1}=0,$ where $k=0,1,2,3,...$ The autor find the ratio between subsequent terms : $$\frac{b_{k}}{b_{k-1}}=-\frac{\gamma_k}{\tilde{\beta_k}-\lambda+\alpha_k\bigg(\frac{b_{k+1}}{b_{k}}\bigg)},$$and reasons "The series should converge and this conditions gives the possible values for $\lambda$". (Indeed $\alpha_k,\gamma_k$ are constants in the limit $k\rightarrow\infty$, and $\beta_k\rightarrow \infty$, making the ratio term vanish at the limit).
From this he writes:$$\lambda=\beta_1-\frac{\alpha_1\gamma_3}{\beta_3-\lambda-\frac{\alpha_3\gamma_5}{\beta_5-\lambda-\cdots}},$$ Which I cannot find a good explanation somewhere. Besides this equation, he writes it in another form which I again cannot reproduce: $$\beta_j-\frac{\alpha_{j-1}\gamma_j}{\beta_{j-1}-\cdots}\cdots\bigg(\frac{\alpha_{0}\gamma_1}{\beta_0}\bigg)=\frac{\alpha_{j}\gamma_{j+1}}{\beta_{j+1}-\cdots}\frac{\alpha_{j+1}\gamma_{j+2}}{\beta_{j+2}-\cdots}\cdots$$ His references:
[1] Leaver E W 1985 Proc. R. Soc. Lond. A 402 285;
[2] Press W H, Teukolsky S A, Vetterling W T and Flannery B P 1992 Numerical Recipes in C 2nd edn (Cambridge: Cambridge University Press)