Usually the partial numerators of a continued fraction are all 1s.
Has anyone considered the operation where you convolve 1 continued fraction with another, in other words, make a new continued fraction formed by replacing the partial numerators of continued fraction $b$ with the terms of continued fraction $a$ (with $a$ being truncated or repeated as necessary to cover the length of $b$) ? Is there any neat way to express the result of this in terms of the original real numbers ?
I am asking this before I have done any work or by-hand examples of this myself. I will probably run something on the computer or by hand over the weekend and will possibly update if it's interesting.