Is this type of 'cfrac' known $\ldots\frac{a_n}{b_n+}\ldots\frac{a_2}{b_2+}\frac{a_1}{b_1+a_0}$?

Question

I have looked in a few well known books on cfracs but did not encounter anything related to the following quantity:

$$ \ldots\frac{a_n}{b_n+}\ldots\frac{a_2}{b_2+}\frac{a_1}{b_1+a_0}=\ldots\cfrac{a_n}{b_n+\cfrac{a_{n-1}}{b_{n-1}+\ldots+\cfrac{a_2}{b_2+\cfrac{a_1}{b_1+a_0}}}} $$

It has a structure similar to ordinary continued fractions but it goes to the upside and to the left (while ordinary cfrac goes to the downside and to the right).

Q: What is known about such a mathematical object?

Answer 1

Your problem is convergence.

Ordinary continued fractions are well-behaved objects because we can truncate them at various points, getting a sequence of rational numbers, which (at least assuming all coefficients are positive integers) converge to some limit. Extending the truncation has a smaller and smaller effect on the final value.

In your backwards continued fractions, that is pretty much never going to be the case. If we truncate stopping at $b_n$, then for the $n^{\text{th}}$ approximation and the $(n-1)^{\text{th}}$ approximation to both be approximately equal to some value $x$, we want $x \approx b_n + \frac{a_n}{x}$, which is just a quadratic condition on $x$, $a_n$, and $b_n$ that ignores the rest of the continued fraction. Similar behavior happens if we truncate stopping at $a_n$. As a result, we don't get convergence unless $a_n$ and $b_n$ approach some specific relationship involving the limit; in particular, if the $a_n$ and $b_n$ are all integers, we can only converge to $x$ if $x$ is the root of a monic quadratic equation with integer coefficients, and we can only converge to it if $a_n$ and $b_n$ eventually are those integer coefficients.