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

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?

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.