Geodesic
Continued fractions with non-integer numerators
Journal of integer sequences, Tome 20 (2017) no. 1
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 
@article{JIS_2017__20_1_a0,
     author = {Greene,  John and Schmieg,  Jesse},
     title = {Continued fractions with non-integer numerators},
     journal = {Journal of integer sequences},
     year = {2017},
     volume = {20},
     number = {1},
     zbl = {1367.11016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2017__20_1_a0/}
}
TY  - JOUR
AU  - Greene,  John
AU  - Schmieg,  Jesse
TI  - Continued fractions with non-integer numerators
JO  - Journal of integer sequences
PY  - 2017
VL  - 20
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/JIS_2017__20_1_a0/
LA  - en
ID  - JIS_2017__20_1_a0
ER  - 
%0 Journal Article
%A Greene,  John
%A Schmieg,  Jesse
%T Continued fractions with non-integer numerators
%J Journal of integer sequences
%D 2017
%V 20
%N 1
%U http://geodesic.mathdoc.fr/item/JIS_2017__20_1_a0/
%G en
%F JIS_2017__20_1_a0
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/