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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 at 6:34
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