Geodesic
Continued fractions with non-integer numerators
Journal of integer sequences, Tome 20 (2017) no. 1.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Anselm and Weintraub investigated a generalization of classic continued fractions, where the "numerator" 1 is replaced by an arbitrary positive integer. Here, we generalize further to the case of an arbitrary real number $z \ge 1$. We focus mostly on the case where $z$ is rational but not an integer. Extensive attention is given to periodic expansions and expansions for $\sqrt n$, where we note similarities and differences between the case where $z$ is an integer and when $z$ is rational. When $z$ is not an integer, it need no longer be the case that $\sqrt n$ has a periodic expansion. We give several infinite families where periodic expansions of various types exist.
Classification : 11A55
Keywords: continued fraction, linear Diophantine equation, pell's equation 
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     author = {Greene, John and Schmieg, Jesse},
     title = {Continued fractions with non-integer numerators},
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Greene, John; Schmieg, Jesse. Continued fractions with non-integer numerators. Journal of integer sequences, Tome 20 (2017) no. 1. http://geodesic.mathdoc.fr/item/JIS_2017__20_1_a0/