Convergence of Exceptional Sequence of irregual Continued Fraction

Question

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 $B_{n}B_{n-1}^{-1}?$

Answer 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.