# Properties of a continued fraction convolution operation

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.

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I would say "convolve" rather than "convolute". – Michael Hardy Feb 8 '13 at 19:10
Now, I would too. Let me correct that. – Cris Stringfellow Feb 8 '13 at 19:19
this is on ice at the moment, and I do want to investigate this. – Cris Stringfellow Feb 16 '13 at 21:01
did anything ever happen with this? – Alexander Gruber Jun 17 '14 at 6:34