# Convergence of Exceptional Sequence of irregual Continued Fraction

All that follows happens in the complex plane.

Consider a convergent irregular continued fraction $$\mathbb{K}(a_n,b_n) := b_0 + \cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\dots}}$$. In the book by Lorentzen and Waadeland, "Continued Fractions, Vol.1" the authors define general convergence of $$\mathbb{K}(a_n,b_n)$$ as, $$\mathbb{K}(a_n,b_n)$$ converges generally to a limit $$l$$ iff there is a sequence $$(\hat{\omega}_n)_n$$ such that the sequence $$S_n(w):=b_0 + \cfrac{a_1}{b_1+\cfrac{a_2}{b_2+\dots\cfrac{a_n}{b_n+\omega}}}$$ satisifies $$S_n(w_n)\to l$$ as $$n\to\infty$$ for all sequences $$(w_n)_n$$, which satisfy $$\liminf_{n\to\infty} d(w_n,\hat{\omega}_n)>0$$ where $$d$$ is the chordal metric.

I am wondering whether there is a way to learn more about the exceptional sequence $$(\hat{\omega}_n)_n$$, in particular convergence. I am especially interested in the sequence $$(B_n)_n$$ defined via $$B_{n+1} = b_n B_{n} + a_n B_{n-1}$$, $$B_0 = 0$$, $$B_1 = 1$$, which defines via $$B_{n}B_{n-1}^{-1}$$ an exceptional sequence for $$\mathbb{K}(a_n,b_n)$$ if $$\mathbb{K}(a_n,b_n)$$ converges. Can I say anything about the limiting behaviour of $$B_{n}B_{n-1}^{-1}?$$

You can learn some asymptotic properties if you know the solution of the continued fraction. Assume that $$S_n(0)$$ converges to some value $$\tau$$. Then for any $$q\neq \tau,$$ the sequence $$\eta_n =S_n^{-1}(q)$$ is a member of the equivalence class of exceptional sequences. As $$n \to \infty$$ these sequences will agree in limit.
Note that $$\eta_n =-S_n^{-1}(\infty)$$ will define the sequence you are looking for.